In a number of diverse situations of interest to me (mostly associated with something called the abelian sandpile model), one can define a nonabelian semigroup generated by commuting elements $a_1,\dots,a_n$ and commuting elements $b_1,\dots,b_n$ such that for all $i$, $a_i$ and $b_i$ do not commute but are semigroup inverses, aka quasi-inverses (i.e., $a_ib_ia_i=a_i$ and $b_ia_ib_i=b_i$). I'd like to know more about the structure of the semigroup. In particular, I'd like to know that for every sequence $i_1, \dots, i_r$ with terms in ${1,\dots,n}$, the products $a_{i_1} \dots a_{i_r}$ and $b_{i_1} \dots b_{i_r}$ are semigroup inverses. Are there standard theorems or methods in semigroup theory that would help me? I should mention that for my applications, the elements $a_i$ and $b_i$ are not inverses in the group sense, even though the semigroup is unital; that is, there is an identity element $e$, but we do not have $a_i b_i = e$ or $b_i a_i = e$.

[In the original version of the question I wrote "$b_i=Ua_iU$ ($1 \leq i \leq n$) for some fixed involution $U$" but then Boris Novikov's question made me realize that for the applications of interest to me $U$ belongs to a larger semigroup, so it seemed best to omit $U$ from the statement of the problem for the time being. I also did not state explicitly that the semigroup is nonabelian.]

There is an extant notion of "sandpile semigroups", but I'm pretty sure that the semigroup I'm interested (introduced by Andrea Sportiello and his coworkers) is something different.

See http://jamespropp.org/pseudo.pdf for a (slightly out-of-date) one-page blurb about the questions that motivated the post, concerning the sandpile model, rotor-router model, and divisible sandpile model. I'm hoping that basic theorems from the theory of semigroups will provide a uniform approach to proving regularity of semigroups in all three contexts.