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This paper explores the difficulties of constructing multidimensional discrete variable representation (DVR) basis sets and the strategies that can be used to overcome them. A parameter count shows that the conditions on a DVR basis set cannot be satisfied on most spaces of wave functions. One-dimensional, orthogonal polynomials are an exception, but the Y[sub lm]'s on the sphere only go 3/4 of the way, in a certain sense, toward supplying enough parameters to satisfy the DVR conditions. It is shown that DVR sets involving rotationally invariant subspaces of wave functions on the sphere (consisting of complete subshells only) exist only for small values of the angular momentum cutoff. However, an exploration of DVR sets invariant under subgroups of the full rotation group leads to the discovery of a 12-point DVR set that is invariant under the tetrahedral group, whose grid points are the vertices of an icosahedron. [ABSTRACT FROM AUTHOR]