Is there any standard name for semigroups $S$ in which $xS=Sx$ for all $x\in S$?


Examples of such semigroups are commutative semigroups and Clifford inverse semigroups.



A name for semigroups in which left and right principal ideals coincide
People in factorization theory call a monoid $H$ *normalizing* if $aH = Ha$ for every $a \in H$; see, e.g.,



> 
> A. Geroldinger, *Non-commutative Krull monoids: A divisor-theoretic approach and their arithmetic*, Osaka J. Math. 50 (2013), 503-539.
> 
> 
> 


However, it makes a lot of sense to refer to the same objects as *duo monoids* (as also suggested by Benjamin Steinberg in a [comment to the OP](https://mathoverflow.net/questions/415718#comment-1066373)), in such a way that a unital ring is duo if and only if its multiplicative monoid is duo (a ring, with or without unity, is said to be *duo* if every left or right ideal is in fact a two-sided ideal).


Duo rings have been studied (under this name) at least since the late 1950s. To the best of my knowledge, they were first considered by E.H. Feller in



> 
> *Properties of primary noncommutative rings*, Trans. Amer. Math. Soc. 89 (1958), 79-91.
> 
> 
> 


