That's what we call the "inversion structure" in concessionary adverbial clauses. (SVC*=>CSV)
As/Although they(S) were(V) happy(C), there was something missing.=>Happy(C) as they(S) were(V), there was something missing.
As/Although Tom is poor, he is honest and diligent. => Poor as Tom is, he is honest and diligent.
Note: Longman Dictionary of contemporary English (P68) has this, "as" also means "though".
"Improbable as it seems, it's true." "Tired as I was, I tried to help them." "As popular as he is (=even though he is popular), the President has not been able to get his own way on every issue."
Hope that helps.