  Symmetry-protected topological crystalline insulators (TCIs) have primarily
been characterized by their gapless boundary states. However, in time-reversal-
($\mathcal{T}$-) invariant (helical) 3D TCI$\unicode{x2014}$termed higher-order
TCIs (HOTIs)$\unicode{x2014}$the boundary signatures can manifest as a
sample-dependent network of 1D hinge states. We here introduce nested
spin-resolved Wilson loops and layer constructions as tools to characterize the
intrinsic bulk topological properties of spinful 3D insulators. We discover
that helical HOTIs realize one of three spin-resolved phases with distinct
responses that are quantitatively robust to large deformations of the bulk
spin-orbital texture: 3D quantum spin Hall insulators (QSHIs), "spin-Weyl"
semimetals, and $\mathcal{T}$-doubled axion insulator (T-DAXI) states with
nontrivial partial axion angles indicative of a 3D spin-magnetoelectric bulk
response and half-quantized 2D TI surface states originating from a partial
parity anomaly. Using ab-initio calculations, we demonstrate that
$\beta$-MoTe$_2$ realizes a spin-Weyl state and that $\alpha$-BiBr hosts both
3D QSHI and T-DAXI regimes.
