  We present a novel class of topological insulators, termed the Takagi
topological insulators (TTIs), which is protected by the sublattice symmetry
and spacetime inversion ($\mathcal P\mathcal T$) symmetry. The required
symmetries for the TTIs can be realized on any bipartite lattice where the
inversion exchanges sublattices. The protecting symmetries lead to the
classifying space of Hamiltonians being unitary symmetric matrices, and
therefore Takagi's factorization can be performed. Particularly, the global
Takagi's factorization can (cannot) be done on a $3$D ($2$D) sphere. In 3D,
there is a $\mathbb{Z}_2$ topological invariant corresponding to the parity of
the winding number of Takagi's unitary-matrix factor over the entire Brillouin
zone, where the $\mathbb Z_2$ nature comes from the $O(N)$ gauge degrees of
freedom in Takagi's factorization. In 2D, the obstruction for a global Takagi's
factorization is characterized by another $\mathbb{Z}_2$ topological invariant,
equivalent to the second Stiefel-Whitney number. For the third-order
topological phases, the $3$D TTIs feature a parity condition for corner
zero-modes, i.e., there always exist odd $\mathcal P\mathcal T$ pairs of
corners with zero-modes. Moreover, for any $\mathcal P\mathcal T$ invariant
sample geometry, all configurations of corner zero-modes satisfying the parity
condition can exist with the same nontrivial bulk topological invariant.
Actually, without closing the bulk gap, the boundary phase diagram have a
cellular structure, where each topological boundary phase associated with a
particular (cross-order) boundary-mode pattern corresponds to a contractible
cell with certain dimension in the parameter space.
