  This work revolves around properties and applications of functions whose
nonlocal gradient, or more precisely, finite-horizon fractional gradient,
vanishes. Surprisingly, in contrast to the classical local theory, we show that
this class forms an infinite-dimensional vector space. Our main result
characterizes the functions with zero nonlocal gradient in terms of two simple
features, namely, their values in a layer around the boundary and their
average. The proof exploits recent progress in the solution theory of
boundary-value problems with pseudo-differential operators. We complement these
findings with a discussion of the regularity properties of such functions and
give illustrative examples. Regarding applications, we provide several useful
technical tools for working with nonlocal Sobolev spaces when the common
complementary-value conditions are dropped. Among these, are new nonlocal
Poincar\'e inequalities and compactness statements, which are obtained after
factoring out functions with vanishing nonlocal gradient. Following a
variational approach, we exploit the previous findings to study a class of
nonlocal partial differential equations subject to natural boundary conditions,
in particular, nonlocal Neumann-type problems. Our analysis includes a proof of
well-posedness and a rigorous link with their classical local counterparts via
$\Gamma$-convergence as the fractional parameter tends to 1.
