The sentence should read They invited you and me, or They invited me and you—either is correct.
A complete sentence consists of a subject, a verb and an object (quite often the object is unseen and unheard, but is understood by the reader or the listener). An example would be, “He is almost as handsome as I.” In that sentence the last word is am, as in “He is almost as handsome as I am.” The am is unseen but is understood.
In the title sentence above, you and I is a compound object that takes the action of the verb invited. This is a very common mistake, one that can easily be avoided by a simple—extremely simple—nay, stupidly simple—process.
There are no complicated rules of grammar to learn. To determine the rightness or wrongness of the sentence, simply delete each of those invited—you and I— in turn, then read the sentence and listen to the sound.
Delete the I and the sentence reads, They invited you, an obviously correct sentence.
Delete the you and the sentence reads, They invited I, an obviously incorrect sentence.
The same simple process may be used when the sentence involves plurals of personal pronouns such as we and us. Were my brother and I conversing and I said, He is almost as handsome as we, the unseen and unheard word would be are, as in we are. We would not say He is almost as handsome as us are.
Special note: The statements regarding relative handsomeness are not necessarily true.
I said it was a simple process–-need I say more?