The problem of checking certain controllability properties of even very simple piecewise linear systems is known to be undecidable. This paper focuses on conewise linear systems, i.e. systems for which the state space is partitioned into conical regions and a linear dynamics is active on each of these regions. For this class of systems, we present algebraic necessary and sufficient conditions for controllability. We also show that the classical results of controllability of linear systems and input-constrained linear systems can be recovered from our main result. Our treatment employs tools both from geometric control theory and mathematical programming.