  We characterize the monomial ideals $I\subset K[x_1,\ldots,x_n]$ with the
property that the polarization $I^p$ and $I^{\sigma^n}:=$ the ideal obtained
from $I$ by the $n$-th iterated squarefree operator $\sigma$ are isomorphic via
a permutation of variables. We give several methods to construct such ideals.
We also compare the depth and sdepth of $I$ and $I^{\sigma^n}$.
