We propose necessary and sufficient conditions for a sensing matrix to be “s-semigood” – to allow for exact ℓ 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 ℓ 1-recovery in terms of the characteristics underlying these conditions. These characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse ℓ 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.