  We introduce the subsystem symmetry-preserving real-space entanglement
renormalization group and apply it to study bifurcating flows generated by
linear and fractal subsystem symmetry-protected topological phases in two
spatial dimensions. We classify all bifurcating fixed points that are given by
subsystem symmetric cluster states with two qubits per unit cell. In
particular, we find that the square lattice cluster state is a
quotient-bifurcating fixed point, while the cluster states derived from
Yoshida's first order fractal spin liquid models are self-bifurcating fixed
points. We discuss the relevance of bifurcating subsystem symmetry-preserving
renormalization group fixed points for the classification and equivalence of
subsystem symmetry-protected topological phases.
