It seems to be an orthographic convention that smooth stops followed by aspirated vowels become rough, e.g., ἔπειτ’ ὁ πατήρ would become ἔπειθ’ ὁ πατήρ. I assume that this is because in the classical pronunciation the former would sound just like the second. But why is there no similar rule in the opposite direction? Why does εἴθ’ ἐξεκόπην not become εἴθ’ ἑξεκόπην?
I assume you mean: “Why does εἴθ’ ἐξεκόπην not become εἴτ’ ἑξεκόπην?”.
The best I (or anyone else) can do is to give you a tautologous answer: because the Greeks pronounced φ, θ and χ as aspirated consonants regardless whether these sounds occurred before a rough or smooth breathing. There’s no reason why they should be pronounced differently when followed by a word beginning with a smooth breathing. On the other hand, there is a reason why π, τ and κ should acquire aspiration and be pronounced as φ, θ and χ when followed by a rough breathing, namely, when joined to the following aspirated vowel, they too acquired aspiration.
There’s orthographic convention, and there’s phonetic reality. We write επειτ(α) ο πατηρ as επειθ’ ὁ πατηρ, with double aspiration. (Whereas we write εἴθ(ε) ἐξεκόπην with single: I take it this was Laurentius’ point.) I don’t imagine that’s phonetically accurate. There’s either aspiration or there’s not, and in these cases there is.
My not so educated guess is that in the case of ἔπειθ’ ὁ πατήρ, we are actually continuing the ancient spelling: ΕΠΕΙΘΟΠΑΤΗΡ. The ancient spelling of εἴθ’ ἑξεκόπην didn’t include any breathing marks, so there is no reason to introduce a convention of introducing a rough breathing after an aspirated plosive.