Associative but NOT Commutative Reducing Function Construction with Existing Hash Function(s)
I don't believe that any such function can meet the requirements of associativity, pre-image resistance and the 'hard to find commuting operands' simultaneously.

Here's why: suppose we have a preimage $a$; we first check if $H(a,a)=a$.  If it is, we found a preimage.

If $H(a,a) = b \ne a$, then we have $H(a,b) = H(a,H(a,a)) = H(H(a,a),a) = H(b,a)$ (because of associativity), and hence we found a commuting pair.