  We present a novel 3D topological insulator, termed the Takagi topological
insulator (TTI), which is protected by the sublattice symmetry and spacetime
inversion symmetry. The symmetries enable the Takagi factorization in the
Hamiltonian space. Due to the intrinsic O(N) gauge symmetry in the Takagi
factorization, a Z2 topological invariant is formulated. We examine the
physical consequences of the topological invariant through a Dirac model, which
exhibits exotic bulk boundary correspondence. The most stable phases are a
number of novel third-order topological insulators featured with odd inversion
pairs of corners hosting zero-modes. Furthermore, the nontrivial bulk invariant
corresponds to a rich cross-boundary-order phase diagram with a hierarchical
cellular structure. Each cell with its own dimensionality corresponds to a
certain configuration of boundary states, which could be of mixed orders.
