Touchstone kullandıımız derslerde bu tür konuşma ağırlıklı yapıları incelemiştik..

so + auxiliary gibi.. as + auxiliary verb kullanılmakta.. belirttiğiniz gibi inversion ama kısmi bir inversion..

alıntıdır;

Partial inversion

In this type of inversion the subject and the auxiliary of the sentence change position. This happens because certain words are emphasised by being placed at the beginning of the sentence. In case of present and past simple the auxiliary do/did is put before the subject and the verb is used in its simple infinitive form. Partial inversion actually looks like a question form, or rather we should say that questions are partial inversions.

.../... (cut)

Partial inversion with AS, SO, NEITHER, NOR + auxiliary + subject to express agreement

Halliburton supported the president’s campaign, as did all the other big companies of corporate America.
"I love Icelandic films." "Really? So do I."
"George can eat fifty eggs in an hour!" "So can my brother!"
"I am not going to Andy’s party tonight." "Nor/Neither is Susan."
"My family have never been on a holiday abroad." "Nor have mine."

Note that if we express an opposing view, there is no inversion:
"I love Icelandic films." "Really? I don’t."