  Alignment is a geometric relation between pairs of Weyl-Heisenberg SICs, one
in dimension $d$ and another in dimension $d(d-2)$, manifesting a well-founded
conjecture about a number-theoretical connection between the SICs. In this
paper, we prove that if $d$ is even, the SIC in dimension $d(d-2)$ of an
aligned pair can be partitioned into $(d-2)^2$ tight $d^2$-frames of rank
$d(d-1)/2$ and, alternatively, into $d^2$ tight $(d-2)^2$-frames of rank
$(d-1)(d-2)/2$. The corresponding result for odd $d$ is already known, but the
proof for odd $d$ relies on results which are not available for even $d$. We
develop methods that allow us to overcome this issue. In addition, we provide a
relatively detailed study of parity operators in the Clifford group,
emphasizing differences in the theory of parity operators in even and odd
dimensions and discussing consequences due to such differences. In a final
section, we study implications of alignment for the symmetry of the SIC.
