  We extend the notion of anticoherent spin states to anticoherent subspaces.
An anticoherent subspace of order t, is a subspace whose unit vectors are all
anticoherent states of order t. We use Klein's description of algebras of
polynomials which are invariant under finite subgroups of SU(2) to provide
constructions of anticoherent subspaces. Furthermore, we show a connection
between the existence of these subspaces and the properties of the higher-rank
numerical range for a set of spin observables. We also note that these
constructions give us subspaces of spin states all of whose unit vectors have
Majorana representations which are spherical designs of order at least t.
