  A Hadamard matrix is balanced splittable if some subset of its rows has the
property that the dot product of every two distinct columns takes at most two
values. This definition was introduced by Kharaghani and Suda in 2019, although
equivalent formulations have been previously studied using different
terminology. We collate previous results phrased in terms of balanced
splittable Hadamard matrices, real flat equiangular tight frames, spherical
two-distance sets, and two-distance tight frames. We use combinatorial analysis
to restrict the parameters of a balanced splittable Hadamard matrix to lie in
one of several classes, and obtain strong new constraints on their mutual
relationships. An important consideration in determining these classes is
whether the strongly regular graph associated with the balanced splittable
Hadamard matrix is primitive or imprimitive. We construct new infinite families
of balanced splittable Hadamard matrices in both the primitive and imprimitive
cases. A rich source of examples is provided by packings of partial difference
sets in elementary abelian 2-groups, from which we construct Hadamard matrices
admitting a row decomposition so that the balanced splittable property holds
simultaneously with respect to every union of the submatrices of the
decomposition.
