Irregular (quasi)varieties of groupoids are (quasi)varieties that do not contain semilattices. The regularization of a (strongly) irregular variety $𝒱$ of groupoids is the smallest variety containing $𝒱$ and the variety $𝒮$ of semilattices. Its quasiregularization is the smallest quasivariety containing $𝒱$ and $𝒮$. In an earlier paper the authors described the lattice of quasivarieties of cancellative commutative binary modes, i.e. idempotent commutative and entropic (or medial) groupoids. They are all irregular...