  We consider symmetry-protected topological (SPT) phases in 2D protected by
linear subsystem symmetries, i.e. those that act along rigid lines. There is a
distinction between a "strong" subsystem SPT phase, and a "weak" one, which is
composed of decoupled 1D SPTs with global symmetries. We propose a natural
definition for strong equivalence of such phases, in terms of a
linearly-symmetric local unitary transformation, under which a weak subsystem
SPT is equivalent to the trivial phase. This leads to a number of distinct
equivalence classes of strong subsystem SPTs, which we show are in one-to-one
correspondence with elements of the group $\mathcal{C}[G_s] =
\mathcal{H}^{2}[G_s^2,U(1)]/(\mathcal{H}^{2}[G_s,U(1)])^3$, where $G_s$ is the
finite abelian onsite symmetry group. We also show that strong subsystem SPTs
by our classification necessarily exhibit a spurious topological entanglement
entropy on a cylinder.
