Let Acom denote the class of aperiodic monoids with commuting idempotents. It is shown that any subvariety of Acom that satisfies the identity x2yx ≈ xyx2 is Cross if and only if it excludes the almost Cross varieties J1, J2, and L. Consequently, these three almost Cross varieties are unique among all subvarieties of Acom that satisfy the identity x2yx ≈ xyx2. On the other hand, the existence of almost Cross subvarieties of Acom different from J1, J2, and L is also established.