We propose necessary and sufficient conditions for a sensing matrix to be “$s$-semigood” — to allow for exact $\ell_1$-recovery of sparse signals with at most $s$ nonzero entries under sign restrictions on part of the entries. We express error bounds for imperfect $\ell_1$-recovery in terms of the characteristics underlying these conditions. These characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse $\ell_1$-recovery and thus efficiently computable upper bounds on those $s$ for which a given sensing matrix is $s$-semigood. We examine the properties of proposed verifiable sufficient conditions, describe their limits of performance and provide numerical examples comparing them with other verifiable conditions from the literature.