Unsplit phrasal verbs with two particles?

In English, the following phrase would sound unnatural:

• */? You can change the brightness settings, adjust the volume, and turn on or off subtitles.

However, if we split the phrase at the end, it sounds better:

• You can change the brightness settings, adjust the volume, and turn subtitles on or off.

Here's another example:

• */? Turn on and off the TV with the remote.

• Turn the TV on and off with the remote.

I can't find a specific rule explaining why we would usually postpose the particles in this case. Since phrasal verbs using multiple particles are kind of rare, there aren't many situations where it even becomes an issue. But I currently have to deal with someone insisting that we should write sentences like "Turn up and down the volume of the speakers."

Heavy (long and complicated) dependents are more likely to appear later in a clause. Hence,

We can turn the TV on or off.

but,

It was the Conservative Government's idea that the way to control the situation and deal with the problem was to turn on and off the tap of consumer demand that was at the root of their stop-go policy. (British Parliament; House of Commons; Prices and incomes; 25 October 1966)

Then again, the construction with on and off between the verb and a relatively light object is not quite so uncommon or unwieldy as assumed in the question. In fact, there are examples to be found which do not offend my ear.

The decision to turn on and off the tap does not rest in one person's hands: That is what is different about individual donations:(HANSARD; House of Commons; Mr Martin Linton; 13 March 2000)

She could turn on and off the TV set with the stubby tip of her middle toe, change channels, even tweak the reception (except for Channel 46, which didn't come in clearly since her father threw his shoe at Dan Rather). (NOVEL: Iris; Buttner, Brenda Lee; 1995)

The hidden units in turn try to turn on or off the output units, obeying the same principles. (Wet Mind: The New Cognitive Neuroscience; Stephen Michael Kosslyn)

In the NATMAP program, a certain parameter allows the support price to be raised or lowered, and another parameter can entirely turn on or off the U.S. dairy price support program. (North American Trade Model for Animal Products; William F. Hahn)

The sheer frequency of the construction with on or off before the object - 831 hits for TURN on or off the in the iWeb Corpus - suggests that it's quite acceptable and that any rule against it would fall under 'style advice' rather than grammatical rule.

The strange thing is that the order V Adv DO is idiomatic only when the DO (direct object) is heavy or the Adv is a single particle.

Admittedly, as pointed out in another answer, people do sometimes write phrases such as "turn on or off the tap". Yes, it's used, but, if the DO is light, people are far more likely to put it next to the verb. We are far more likely to "turn the computer off and on" than to "turn off and on the computer".

Yes, some confouding factors exist, but, to avoid them, consider sentences where the DO is a light noun phrase. (The DO can't be just a pronoun; if it is, it must go before the other argument.) And consider the question not whether or not a phrase is attested at all but whether it is idiomatic. On this basis I would exclude "turn on and off the TV".

So, then, what can go in the Adv position even if the DO is light, the resulting pharse being idiomatic?

It's curious. Some adverbs work, such as on, off, up, down, as the OP and others have noted. Other examples:

• Don't let in the cold.

• Put out the candles.

They work whether the particle has no independent meaning, has a figurative meaning, or bears its usual meaning.

• Lock up your valuables.

• Turn up the volume.

• Lift up the lid.

Typical adverbs used in this way make for unidiomatic phrases:

• *I turned sideways the picture.

• *I visited recently/yesterday London.

• *I read quickly/fast the book.

• *I watched excitedly the film.

The fact that each of the particles which work is a homonym of a preposition might suggest that it is a preposition, and that the pattern here is that prepositions work but adverbs don't. One trouble with this is that prepositions can be conjoined, and the result can be used the way a single preposition can:

• I walked up and down the path.

whereas, as has been noted, such phrases as "turn up and down the volume" are rare and unidiomatic. Another is that, with some "prepositions", the word order with DO last doesn't work:

• I lifted the flap and squeezed the package through.

• *I lifted the flap and squeezed through the package.

And at least one adverb that does work is not also used as a preposition:

• That brought back some memories.

But I have difficulty in finding where to draw the line. These last examples bring home the difficulty.

• ?I brought home the shopping.

• I bring home the bacon.

I tentatively suggest that there is a lightness constraint on the adverb. A test failed by coordinations such as "off and on", and by most adverbs, but passed by a few adverbs. The latter are short. But the matter of whether or not the word can also be used as a preposition is not a reliable guide to whether or not it works in this context.