In this paper, we investigate some new interesting solution structures of the (2+1)-dimensional bidirectional Sawada–Kotera (bSK) equation. We obtain soliton molecules by introducing velocity resonance. On the basis of soliton molecules, asymmetric solitons are obtained by changing the distance between two solitons of molecules. Based on the N-soliton solutions, several novel types of mixed solutions are generated, which include the mixed breather-soliton molecule solution by the module resonance of the wave number and partial velocity resonance, the mixed lump-soliton molecule solution obtained by partial velocity resonance and partial long wave limits, and the mixed solutions composed of soliton molecules (asymmetric solitons), lump waves, and breather waves. Some plots are presented to clearly illustrate the dynamic features of these solutions. Copyright © 2020 Chinese Physical Society and IOP Publishing Ltd.