  We study the classification of symmetry protected topological (SPT) phases
with crystalline symmetry (cSPT phases). Focusing on bosonic cSPT phases in two
and three dimensions, we introduce a simple family of cSPT states, where the
system is comprised of decoupled lower-dimensional building blocks that are
themselves SPT states. We introduce a procedure to classify these block states,
which surprisingly reproduces a classification of cSPT phases recently obtained
by Thorngren and Else using very different methods, for all wallpaper and space
groups. The explicit constructions underlying our results clarify the physical
properties of the phases classified by Thorngren and Else, and expose
additional structure in the classification. Moreover, the states we classify
can be completely characterized by point group SPT (pgSPT) invariants and
related weak pgSPT invariants that we introduce. In many cases, the weak
invariants can be visualized in terms of translation-symmetric stacking of
lower-dimensional pgSPT states. We apply our classification to propose a
Lieb-Shultz-Mattis type constraint for two-dimensional spin systems with only
crystalline symmetry, and establish this constraint by a dimensional reduction
argument. Finally, the surprising matching with the Thorngren-Else
classification leads us to conjecture that all SPT phases protected only by
crystalline symmetry can be built from lower-dimensional blocks of invertible
topological states. We argue that this conjecture holds if we make a certain
physically reasonable but unproven assumption.
