grammar - "I and Jane" or "me and Jane"?

So I know that it's correct to say:

Jane and I are going shopping

I shouldn't use me here because (as stated on Oxford Dictionaries Online) I is what I would use in the singular form of the sentence:

I am going shopping

But does this same rule apply if I want to (impolitely) put myself first in the sentence? Should I be saying:

I and Jane are going shopping

Because this just sounds incredibly wrong. It sounds much more accurate to use me here.

Answer

There are very strong indications that " x and me" or "me and x" are the actual grammatically-correct form in English (in concordance with French and with historical usage) and the horrible resonance of "I and x" is among them.

The problem with the "me" formulation is that it will never get by an editor or reader who has taken primary school English, except in a quotative construction relating non-standard speech. The influence of classically-educated grammarians runs deep, and the idea that subject pronouns must appear in subject positions has been so thoroughly drilled into people that "x and I" sounds right to most people, even though it is arguably wrong according to the natural grammar of the language. (If one substitutes "a group consisting of [members list]", "me" is the proper constituent.)

The upshot is that if you want to be taken seriously in a non-colloquial context, you pretty much have to use "x and I". "Me and x" or "x and me" will sound too colloquial or sloppy, and "I and x" simply sounds wrong to most folk.