Summary: In this article we continue the investigation of insertion-deletion systems having a context only on one side of insertion or deletion rules. We show a counterpart of the results obtained in [{\it A. Matveevici}, {\it Y. Rogozhin} and {\it S. Verlan}, “Insertion-deletion systems with one-sided contexts", Lect. Notes Comput. Sci. 4664, 205‒217 (2007; Zbl 1155.68419)] by considering corresponding systems and exchanging deletion and insertion parameters. We prove three computational completeness results and one non-completeness result for these systems. We also solve the remaining open problem concerning the generative power of insertion-deletion systems having both contexts by proving the computational completeness of systems having a context-free insertion of two symbols and a contextual deletion of one symbol.