We present a numerical investigation of the modes of adhesion and endocytosis of two spherocylindrical nanoparticles (SCNPs) on planar and tensionless lipid membranes, using systematic molecular dynamics simulations of an implicit-solvent model, with varying values of the SCNPs' adhesion strength and dimensions. We found that at weak values of the adhesion energy per unit of area, ξ, the SCNPs are monomeric and adhere to the membrane in the parallel mode. As ξ is slightly increased, the SCNPs dimerize into wedged dimers, with an obtuse angle between their major axes that decreases with increasing ξ. However, as ξ is further increased, we found that the final adhesion state of the two SCNPs is strongly affected by the initial distance, d0, between their centers of mass, upon their adhesion. Namely, the SCNPs dimerize into wedged dimers, with an acute angle between their major axes, if d0 is relatively small. However, for relatively high d0, they adhere individually to the membrane in the monomeric normal mode. For even higher values of ξ and small values of d0, the SCNPs cluster into tubular dimers. However, they remain monomeric if d0 is high. Finally, the SCNPs endocytose either as a tubular dimer, if d0 is low or as monomers for large d0, with the onset value of ξ of dimeric endocytosis being lower than that of monomeric endocytosis. Dimeric endocytosis requires that the SCNPs adhere simultaneously at nearby locations.