  We introduce a family of highly symmetric bipartite quantum states in
arbitrary dimensions. It consists of all states that are invariant under local
phase rotations and local cyclic permutations of the basis. We solve the
separability problem for a subspace of these states and show that a sizable
part of the family is bound entangled. We also calculate some of the Schmidt
numbers for the family in $d = 3$, thereby characterizing the dimensionality of
entanglement. Our results allow us to estimate entanglement properties of
arbitrary states, as general states can be symmetrized to the considered family
by local operations.
