The time reversal and chiral symmetry are special because they are antiunitary symmetries, in contrast to the other unitary symmetries like translation and rotation symmetries. Antiunitary symmetry operation involves complex conjugation of the wave function of the system, which is a non-trivial operation beyond unitary transforms.
Unitary symmetries are simpler to deal with because the symmetry operation $U$ commutes with the Hamiltonian $H$ (as $UHU^\dagger=H$), which means the Hamiltonian can be block-diagonalized in the irreducible representations of the symmetry group. Within each representation, one can just study the small block-Hamiltonian labeled by the symmetry quantum number. For example, if the system has translation symmetry, the Hamiltonian can be block-diagonalized by Fourier transform to the momentum space. Each block-Hamiltonian is labeled by the momentum quantum number. Then one can further study the effect of other symmetry operations in each momentum block.
However with antiunitary symmetries, such block diagonalization is no longer possible. Different states connected by time-reversal symmetry should be put together and analyzed as a whole. There no separation of representations by symmetry quantum numbers for antiunitary symmetries, which make the antiunitary symmetries non-trivial and harder to handle.
When it comes to the classification of symmetry protected topological states in condensed matter physics, it is usually assumed that the unitary symmetries have been treated by transforming the Hamiltonian to the block-diagonal basis of their irreducible representations. Then the antiunitary symmetries are the only symmetries that are left for more detailed case-by-case analysis, which eventually leads to the famous ten-fold-way classification (http://arxiv.org/abs/0912.2157).