On polarised class groups of orders in quartic CM-fields

We give an explicit characterisation of pairs of orders in a quartic CM-field that admit the same polarised ideal class group structure. This generalises a simpler result for imaginary quadratic fields. We give applications to computing endomorphism rings of abelian surfaces over finite fields, and extending a completeness result of Murabayashi and Umegaki [13] to a list of abelian surfaces over the rationals with complex multiplication by arbitrary orders.