In this manuscript, we present some new results for the semidefinite linear complementarity problem in the context of three notions for linear transformations, viz., pseudo $w-P$ property, pseudo Jordan $w-P$ property, and pseudo SSM property. Interconnections with the $P_#$-property (proposed recently in the literature) are presented. We also study the $R_#$-property of a linear transformation, extending the rather well known notion of an $R_0$-matrix. In particular, results are presented for the Lyapunov, Stein, and the multiplicative transformations.