Why is "herself" required in this particular sentence?

Solution 1:

Here are the three cases you have presented:

(1) I gave Susie a picture of herself.

(2) I went there by myself.

(3) Do it yourself.

I have come up with a new rule under which your three cases as well as the traditional rule are subsumed:

A personal pronoun must be in the form of a reflexive pronoun in order to refer back to another word when both the word and the personal pronoun are contained within a single clause.

I'll call this new rule "the single-clause rule". The single-clause rule surely incorporates the traditional "subject-object" rule, because a subject and an object are contained in a single clause.

Now, let's see if the single-clause rule explains the three cases above.

Regarding (1) and (2), herself and myself refer back to Susie and I, respectively. In (3), yourself refers back to the implied subject you. And these are all in a single clause.

Solution 2:

Okay, so to be free of that 400 character restriction.

A reflexive pronoun is defined as:

Reflexive pronouns are pronouns that refer back to the subject of the sentence or clause. They either end in –self, as in the singular form, or –selves as in the plural form.

But they don't always have to be the subject and I have searched for a few minutes to find anything that didn't have the "subject" requirement. But this definition is partly true. Most of the time reflexive pronouns refer back to the subject of the sentence. Not always the case. I would say that a reflexive pronoun refers back to a specific individual within a sentence when, if the reflexive is left out, it would lead to ambiguity. Take the the following picture example. In it, saying "I gave Susie a picture of her" could refer to someone else. "Her" could refer to Susie's daughter or sister or friend or possibly a shared friend of the speaker/writer. "Her" could be substituted with "him" and we would only have a change of sex.

As you've stated yourself, there is a difference in meaning between

I gave Susie a picture of her.

And:

I gave Susie a picture of herself.

The first means that Susie was given a picture of another girl whereas the second means that Susie received a picture of herself (Susie).

Similarly:

He hit him.

would mean Person A ["John"/"Joe"/...] hit Person B ["Billy"/"Bob"] while

He hit himself.

would mean Person A hit Person A [=himself].

Now it doesn't have to be a pronoun in subject position either:

John hit himself. [≠ John hit him.]

The meanings are drastically different. In the first, John is harming his body; in the second, John is harming someone else (someone unknown to the reader).

I went there by myself.

means I went there alone while by me doesn't even make sense.

But the Do it yourself, yourself seems like a variant of You do it yourself with the initial emphatic yourself moved to the end?

First, are you asking a question here? Second, I don't see the need for the second yourself when the first emphasizes it just fine. Third, the sentence is an imperative, which means that the implied subject is you; thus, changing yourself to you and then placing it at the beginning of the sentence make me think that it is used to address someone (e.g., John, come here).

What I'm really interested in though is what the grammatical rules are which dictate that the reflexive must be used in I gave Susie a picture of herself for example.

I'm not sure it's so much about grammar as it is about the semantics(?). The use of reflexive/intensive pronouns is when we want to avoid ambiguity; take the first example. Although the only antecedent to the pronoun her is Susie, the pronoun could be referring to a previously mentioned individual (e.g., Mary), but the reflexive pronoun herself eliminates the ambiguity by making it a direct reference to that mentioned individual.

I note in passing that yourself seems to be an adjunct ('adverbial') here.

It is not an adjunct here. It is an intensive pronoun:

The intensive pronouns (also called emphatic pronouns) are: myself, yourself, herself, himself, itself, ourselves, yourselves, and themselves.

These words can be either intensive pronouns or reflexive pronouns.

Here are some examples of intensive pronouns: 1. She will do it herself. (The intensive pronoun herself emphasizes that she will do it. Her husband won't do it. Her son won't do it. SHE will do it.)

The boys baked these scones themselves. (The intensive pronoun themselves emphasizes that the boys baked the scones, i.e., not their mothers.)

I heard the lie myself. (The intensive pronoun myself emphasizes that I heard the lie.)

The dog opened the cupboard itself.

You can test if it's an intensive pronoun by removing it and seeing if you get the same effect by emphasizing the thing you're trying to emphasize with your voice (shown here in uppercase).

SHE will do it.

I heard the lie.

THE DOG opened the cupboard.

I added this to provide more direct information on intensive pronouns.

I originally gave the imperative example for two reasons. Firstly, the reflexive pronoun here is obviously not the object of the verb. Secondly, however, I gave it because there is no antecedent word in this sentence which could be the subject of the verb, or, which represents the same person as yourself.

Yourself does have an antecedent: it's you. Imperative sentences have an understood you as subject. So yourself references the subject.

Pass the salt.

Shut the door.

Smile.

What is the subject of those sentences? Hmm… that’s tricky!

This may sound strange, but every single command has the same subject! Yikes! How is that even possible?

Well, since commands are always speaking to someone or something (you’ve got to address them if you’re going to ask them to do something), the subject is always the word you.

You may have noticed that the word “you” is not even in a command. Because of this, the subject is actually called you understood, and it is written like this: (you).

This means that the subject is the word you, but since you is not written or spoken in the sentence, it is simply understood and is written in parentheses.

Solution 3:

I gave you a picture of me

and the less grammatically correct but idiomatically common

I gave you a picture of myself

i.e., I gave you a picture of Mari-lou. It is a photo of me/myself, I am in that photo.

Likewise, the phrase I gave Susie a picture of herself means that the photo is of Susie's self; i.e., herself. She is in that photo.

You could say

I gave Susie a picture of Susie.

That works, although repetitive, it is grammatical. Susie is a relatively uncommon name so we would guess that the photo is an image of the same person but it might be that there are two Susies, who can tell? The reflexive pronoun herself removes the ambiguity.

Compare

I gave Susie a picture of her

Her could refer to Susie or to any woman by using the reflexive pronoun, herself, we understand specifically who is in the photo.

I gave Susie a picture of me or myself (a photo of me)
...... herself (a photo of Susie)
...... her (any woman)
...... him (any man)
...... us (no ambiguity, it is a photo of Susie and me)
...... them (two or more people).

Exceptions

In some languages there are verbs that require the reflexive pronoun that are omitted in English. Examples are shave, wash, and dress. In Italian (and I believe in Spanish too) the subject performs these actions on him or herself, thus we have:

radersi to shave (oneself)

mi rado = I shave(myself)
ti radi = you shave (yourself)
si rade = he/she shaves (him/herself)
vi radete = (pl) you shave (yourselves) etc.

lavarsi to wash (oneself)

mi lavo = I wash (myself)
you lavi = you wash (yourself)
si lava = he/she washes (him/herself)
vi lavate = you wash (yourselves)

The British Council website succinctly explains:

NOTE: We do not use a reflexive pronoun after verbs which describe things people usually do for themselves:

1) He washed in cold water.
2) He always shaved before going out in the evening.
3) Michael dressed and got ready for the party.

We only use reflexives with these verbs for emphasis:

He dressed himself in spite of his injuries.
She’s old enough to wash herself.

EDIT
See usage note in Dictionary.com concerning myself vs. me

As part of a compound subject, object, or complement, myself and to a lesser extent the other -self forms are common in informal speech and personal writing, somewhat less common in more formal speech and writing: The manager and myself completed the arrangements. Many came to welcome my husband and myself back to Washington.
[...]
There is ample precedent, going as far back as Chaucer and running through the whole range of British and American literature and other serious formal writing, for all these uses. Many usage guides, however, state that to use myself in any construction in which I or me could be used instead (as My daughter and myself play the flute instead of My daughter and I, or a gift for my husband and myself instead of for my husband and me) is characteristic only of informal speech and that such use ought not to occur in writing.

Solution 4:

Without yourself, "do it" is simply a command to do something. Do it yourself has the connotation that you have implied someone else should do it, and the statement being made is about who should do it, not the fact that it should be done.

Solution 5:

I'm not a crazy grammar genius, but I'm pretty sure it's all self-explanatory.

I gave Susie a picture of her.

This sentence means that Susie was given a picture of [some girl].

I gave Susie a picture of herself.

This sentence means that Susie was given a picture of Susie.

I went by myself.

If you break down this sentence, myself answers the question:

I went by what means?

It's the same structure as if you had said, "I went there by car." And since the means refers back to the subject I, you use reflexive form.

Do it yourself.

I was always taught that when you're using a second-person viewpoint, sentences such as this have an understood subject you as in the sentence:

You do it yourself.

And again, since yourself is referring back to the subject you, you use a reflexive form.