What's the single-word for "left or right handed"?

Another option would be hand-dominance or hand-dominancy as a single word. It is not as common as handedness and I usually see it in medical contexts. (It is usually in two words but you can hyphenate to form a hyphenated compound noun).

Research shows that hand dominancy can be an important feature affecting one's tolerance of pain. According to a 2009 Israeli study from the University of Haifa, right-handed people are more tolerant of pain than left-handed individuals.

_{Understanding Pain: An Introduction for Patients and Caregivers By Naheed Ali, Moshe Lewis}

It might encompass the left-hand-dominance and right-hand-dominance better than handedness. Handedness can be better understood as a broader term that covers mixed-handedness and ambidexterity also.

They are mentioned as synonyms in some contexts but here is an explanation of the differences between hand dominance, handedness and hand preference:

Hand dominance means that one hand has the most influence or control. Handedness means that one hand is more reliable for use across a range of skillful acts. Hand preference means that one hand is preferred or chosen.

_{www.erhardtproducts.com}

Additionally, there are eye-dominance and ear-dominance; and there is the hypernym side-dominance that covers all different types of dominance.

I don't think you'll improve on handedness (wikipedia) (reference.com). It's not a very nice word admittedly, and sounds a bit like what you started with, but there aren't any synonyms listed in this sense.

Handedness is the most common and most understandable term. But I'm going for my pedant merit badge, so I'm going to draw upon and explain the other suggestions.

A history and evolution of meaning: Chiral

@DanBron suggests the technical term Chirality:

The word chirality is derived from the Greek, χειρ (kheir), "hand", a familiar chiral object.

An object or a system is chiral if it is distinguishable from its mirror image; that is, it cannot be superposed onto it.

There has been a great degree of speculation as to whether chirality is applicable to the specific chiral orientation or if it is merely the property of being non-superposable. To clear it up, consider the following definitions.

chiral
adjective, Chemistry
1. (of a molecule) not superimposable on its mirror image.

_{chirality. (n.d.). Collins English Dictionary - Complete & Unabridged 10th Edition. Retrieved April 28, 2015, from Dictionary.com website: http://dictionary.reference.com/browse/chirality}

Chiral is the adjectival property of whether an object possesses symmetry, dissymmetry; whether it is superimposable or it is achiral/amphichiral. These have the same definition: not chiral; lacking any chirality in the same way that asexual means non-gendered; neither male nor female. In chemistry, pairs of molecules with opposite chirality are called enantiomer's, sometimes optical isomers; in mathematics enantiomorphs is more common.

chirality
noun 1. the configuration or handedness (left or right) of an asymmetric, optically active chemical compound. Also called dissymmetry

_{chirality. (n.d.). Collins English Dictionary - Complete & Unabridged 10th Edition. Retrieved April 28, 2015, from Dictionary.com website: http://dictionary.reference.com/browse/chirality}

Chiral is a quite new addition to the English language, having only been though up by Lord Kelvin in 1893. The definitions for the word in technical fields and common language are evolving, but the suffix -ity is winning out for the meaning of chirality.

Used to form a noun from an adjective; especially, to form the noun referring to the state, property, or quality of conforming to the adjective's description.

Handedness vs. Chiral

Handedness is commonly used to distinguish between the distinct items of a chiral pair – indeed, almost every definition or explanation of chirality I've heard (or Google suggested just now) uses the mirror symmetry of human hands as a prime example. For instance in physics, for subatomic particles that have a spin, one will be referred to as having left-handed spin, the other as having right-handed spin. It's also applied to describe helix structures (DNA/RNA and threaded screws), electromagnetic and other vector fields where the right-hand-rule is well known, knots and several other geometric and mathematical notions.

JasonC suggests that handedness is a behavioral preference and therefore chirality is not an applicable synonym. However, manual dexterity is as least much a matter of skill as preference. Becoming a switch hitter in baseball requires a great deal of work. Many highly trained and motivated professional athletes do not succeed at overcoming this behavioral preference even though it provides a noticeable advantage to their career. People tend to see their hand dominance (or preference) as a part of themselves, a defining characteristic. In that sense, handedness is as much a physical configuration as a behavioral one.

Chirality is technically correct when applied to people's hands, but almost never used. (That's part of why I like it; it's the best kind of correct.) However, in American English, it's not the right way to ask a person about themselves, mostly because it doesn't communicate the question as several commenters have pointed out. Communication will fail at a rate inversely proportional to the renown of the words (and grammar) used; the words chiral and chirality are not well known by many people without specialized knowledge of science, technology, engineering or math, so don't use chirality. It is not used and will not be understood in common language. If you're pitching a design for a nightlight (or even a horror movie), you can't just say "Are you scotophobic, nyctophobic, lygophobic or achluophobic? Then you'll love this!"

I'm basing my suggestion against chirality and the rest of this answer on the assumption that anyone talking about their designs is seeking to communicate with a broad audience. In short, you question sounds like you're trying to market your designs and marketing messages that are not understood often achieve the opposite of the desired effect. You will drive your clientele away if they don't understand you with minimal effort. If you are specifically trying to be coy and pique their interest by challenging their assumptions, I still suggest you stay away from uncommon technical terms. Unless you are speaking directly and exclusively to an audience you have every reason to expect will know the term chirality, I suggest you don't use it.

Furthermore, chirality might be assumed by people who know the greek root to referred to hand preference. Or a listener might assume you're asking about their hand preference because our left and right hands are the most ubiquitous identifiers of our asymmetry. But technically speaking, chirality could refer to any aspect of human asymmetry, of which there are many: wrists, elbows, shoulders, legs, feet, kidneys, ears etc. etc. all cannot be superimposed on their mirrored pair in the body.

• I lost a foot to a tractor accident.
• My left nostril is slightly larger.
• Every body part I have two of.

Are all correct and applicable answers to the question "What is your chirality?"

But even replacing it with the strict synonym 'handedness' can be confusing.

Eliciting a left-or-right response

I think your first example is best filled by using two words (and to hell with the single-word-request tag). I think it's far more common and understandable to ask:

What's your dominant hand?

This makes it most clear that you're interested in an either-or, left-or-right answer. As some commenters point out, handedness is a spectrum; some people may write with one hand and eat, brush their teeth, cut their food etc. with the other. Nonetheless, when asked directly about which hand is dominant, most people will correctly identify that it's left or right, map it to whatever activities they consider most important, and answer 'left' or 'right'.

_{I'm basing this on the asker's mention of "state of being left- or right-handed" indicating the desire for an either-or, binary answer. If you want a nuanced answer on a spectrum, use the more open ended "handedness" option.}

Inclusive instead of Exclusive language

Similarly, I think it would be very weird to hear:

Our designs do not bias against gender. Our designs are a-gendered/gender neutral.

Our designs do not bias against handedness. (Or, for that matter...)

Our designs do not bias against chirality. Our designs are achiral/chiral neutral.

The distinction itself - gender or handedness - isn't what you'd be biased towards or against. That phrasing implies your designs are for suitable for both gendered and asexual people (or handed and ambidextrous people). I think what you're trying to convey is that your designs are suitable for all people, whether they left-handed, right-handed or ambidextrous. Therefore, I humbly suggest (or rather second Chris H.'s suggestion):

Our designs are ambidextrous.

The prefix ambi- is inclusive, meaning both, unlike a prefix of un- or a- which would mean neither. Saying a person is ambidextrous means they use both hands equally well. Saying an inanimate object or design is ambidextrous implies that people can use either hand to hold, use or operate it equally well. A steering wheel is ambidextrous - it would be hard to imagine one that wasn't (because circles lack chirality). A gear shifter is typically right-handed in cars with a left-handed driver seat (and vice-versa in Britain, e.g.).

And if you're talking about an animate object, ambidextrous is the only way to go!

Our robot designs are ambidextrous.

As you correctly point out, however, you can't remove the prefix from ambidextrous and make any kind of sense. Dexterity by itself doesn't describe handedness or chirality. If you ask someone about their dexterity, you'd get a wide variety of interpretations - my grandmother would start talking about knitting and I'd roll three six-sided dice.

Sliding back into the marketing angle, promoting your product or design as not biased is taking a defensive tone. It is building a subtle connection in the listeners mind between your product and bias. I suggest you try very hard to find a positive or neutral way to describe your designs.

A washbasin with the tap on the center is, then, handedness-blind or chirality-blind, if you will.

It's also precisely ambidextrous.

ambidextrous
adjective
1.1 (Of an implement) designed to be used by left-handed and right-handed people with equal ease:

_{ambidextrous. (n.d.). In Oxford Dictionaries. Retrieved April 29, 2015, from http://oxforddictionaries.com/definition/english/ambidextrous}

I suggest laterality.
I think this works, it's less obscure than chirality; and readers should find the term, laterality, self-explanatory if used in the proper context.

noun; 1640-1650

1. the use of one hand in preference to the other.
   Compare handedness.

laterality in Medicine
laterality lat·er·al·i·ty (lāt'ə-rāl'ĭ-tē)
n. Preferential use of limbs of one side of the body.

2. the dominance or superior development of one side of the body or brain.

Examples of laterality used in literature

• Humans are bilateral animals; accordingly, laterality is the dominance, or preference of, one side of the body or brain, such as the hand or foot (sides usually being left or right).

• There is no extensive agreement on the origin of manual preferences. Some believe such laterality is inherited; others, that the child is trained to it; and still others, that biases are initiated in an infant during pregnancy

• Laterality refers to an important change in the brain that permits an ‘internal awareness of the two sides of the body and their differences’ (Newel Kephart) [...] Until a child is about 3 years of age both sides of the brain perform the same types of functions. [...] The two hemispheres of the brain are still doing the same things in the same way. If your child has had plenty of sensory and motor experiences as an infant and toddler, the brain matures and certain functions are specialised in one hemisphere or side of the brain – this is the process of lateralisation (the word lateralisation’ comes from the Latin word meaning ‘side’).

• Other existing asymmetries or lateralities in humans, such as arm folding, hand clasping, and leg crossing, have been ...

There are two advantages to preferring laterality which the term handedness does not offer. It is a more rigorous expression. And it fits perfectly in the following statement—

Our designs are not biased against your laterality.

I cannot quite tell why you want to know this, but I strongly advise rewording.

You seem to think everybody is one thing or the other, while in fact this has no basis in biology; most people’s handedness exists on a continuum, just as they do with so many other factors, including footedness and eyedness and a whole lot more. Although I am not one of them, I know lots of people whose handedness, footedness, and eyedness are not all three of them the same sort of lateral dominance. For example, they might be left-handed and -footed but right-eyed.

However, if you just want clumping not nuance, you should ask exactly what you mean and not use fancy words or beat around the bush:

1. Are you left-handed?

If you want nuance, then ask about each aspect:

2. What hand do you usually write with?
3. What hand do you usually eat with?
4. What hand do you usually throw with?
5. What hand do you usually catch with?

What is your purpose? I feel like you’re trying to squeeze something into a slot that English doesn’t put there. If you what to know whether someone is right- or left-handed, for goodness’ sake just ask them that! You are making a mistake by trying to formulate a complete sentence and then seeking a single word that fits in that slot. So your sample:

What’s your ____?

Is a problem because that isn’t what people ask. They ask:

• Are you right-handed or left-handed?

Now and then they might ask:

• Which hand do you use?

Sure, you could ask

• What’s your dominant hand?

But that’s a pretty specialized and clinical use. The same goes with eyes and feet, where just like hands, a person tends to be dominant in one but not the other. The right question then is:

• Are you left-eyed?

or even

• Are you left- or right-eye dominant?

Even so, that sounds intrusive. For the most part, this is just not a question that gets asked, and if it is, one asks in the normal way, not in a way that reaches for some abstruse term nobody ever uses. That’s completely counterproductive.

Same with feet. Asking

• What’s your dominant foot?

Just sounds funny, and people do not usually talk that way.

You have the same problem with saying:

Our designs are not biased against your ____.

While it is possible to say

• Our designs are not biased against your manual orientation.

You are apt to confuse all but the quick-witted, again because people really don’t talk about these things this way in English.

I suppose you could say

• Our designs are not biased against left-handers.

Which is what you really mean, but it comes off sounding more than a little sinister, so I wouldn’t trust that not to bother someone or other. Then again, so does everything.