  In the four-dimensional Hilbert space, there exist 16 Heisenberg--Weyl (HW)
covariant symmetric informationally complete positive operator valued measures
(SIC~POVMs) consisting of 256 fiducial states on a single orbit of the Clifford
group. We explore the structure of these SIC~POVMs by studying the symmetry
transformations within a given SIC~POVM and among different SIC~POVMs.
Furthermore, we find 16 additional SIC~POVMs by a regrouping of the 256
fiducial states, and show that they are unitarily equivalent to the original 16
SIC~POVMs by establishing an explicit unitary transformation. We then reveal
the additional structure of these SIC~POVMs when the four-dimensional Hilbert
space is taken as the tensor product of two qubit Hilbert spaces. In
particular, when either the standard product basis or the Bell basis are chosen
as the defining basis of the HW group, in eight of the 16 HW covariant
SIC~POVMs, all fiducial states have the same concurrence of $\sqrt{2/5}$. These
SIC~POVMs are particularly appealing for an experimental implementation, since
all fiducial states can be connected to each other with just local unitary
transformations. In addition, we introduce a concise representation of the
fiducial states with the aid of a suitable tabular arrangement of their
parameters.
