  Using density-matrix renormalization-group calculations for infinite
cylinders, we elucidate the properties of the spin-liquid phase of the
spin-$\frac{1}{2}$ $J_1$-$J_2$ Heisenberg model on the triangular lattice. We
find four distinct ground-states characteristic of a non-chiral, $Z_2$
topologically ordered state with vison and spinon excitations. We shed light on
the interplay of topological ordering and global symmetries in the model by
detecting fractionalization of time-reversal and space-group dihedral
symmetries in the anyonic sectors, which leads to coexistence of symmetry
protected and intrinsic topological order. The anyonic sectors, and information
on the particle statistics, can be characterized by degeneracy patterns and
symmetries of the entanglement spectrum. We demonstrate the ground-states on
finite-width cylinders are short-range correlated and gapped; however some
features in the entanglement spectrum suggest that the system develops gapless
spinon-like edge excitations in the large-width limit.
