How much symmetry do symmetric measurements need for efficient operational applications?

We introduce a generalization of symmetric measurements to collections of unequinumerous positive, operator-valued measures (POVMs). This provides a uniform description of objects that are more general than symmetric, informationally complete POVMs and mutually unbiased bases, but at the same time less destructive and more noise tolerant. For informationally complete sets, we propose construction methods from orthonormal Hermitian operator bases. The correspondence between operator bases and measurements can be as high as one-to-four, with a one-to-one correspondence following only under additional assumptions. Importantly, it turns out that some of the symmetry properties, lost in the process of generalization, can be recovered without fixing the same number of elements for all POVMs. In particular, for a wide class of unequinumerous symmetric measurements that are conical 2-designs, we derive the index of coincidence, entropic uncertainty relations, and separability criteria for bipartite quantum states.