In this work, we study the behavior of electromagnetic complementary composite periodic media via an approach based on their supported eigenmodes for in-plane wave propagation. The numerical eigen-analysis which is based on the spurious-free field-flux eigenmode finite element formulation proves the need for metal-backing of the structure to support the complementary mode. Parametric analyses are presented for the cases of varying distances between the resonators and the ground plane. We also prove the feasibility of complementary structures to support resonance modes with partial ground termination of metallic strips leading to a novel metamaterial particle presented first time in the literature. Experimental results confirm the numerical simulations and prove its potential for incorporation in more complex microwave planar structures.