  Quantized responses are important tools for understanding and characterizing
the universal features of topological phases of matter. In this work, we
consider a class of topological crystalline insulators in $3$D with $C_n$
lattice rotation symmetry along a fixed axis, in addition to either mirror
symmetry or particle-hole symmetry. These insulators can realize quantized
mixed geometry-charge responses. When the surface of these insulators is
gapped, disclinations on the surface carry a fractional charge that is half the
minimal amount that can occur in purely $2$D systems. Similarly, disclination
lines in the bulk carry a fractionally quantized electric polarization. These
effects, and other related phenomena, are captured by a $3$D topological
response term that couples the lattice curvature to the electromagnetic field
strength. Additionally, mirror symmetric insulators with this response can be
smoothly deformed into a higher-order octopole insulator with quantized corner
charges. We also construct a symmetry indicator form for the topological
invariant that describes the quantized response of the mirror symmetric
topological crystalline insulators, and discuss an unusual response
quantization in time-reversal breaking systems.
