We have acquired deep, H-band, imaging for a sample of 286 Virgo cluster galaxies with B <= 16 mag and extracted surface photometry from optical g,r,i,z Sloan Digital Sky Survey images of 742 Virgo Cluster Catalog galaxies, including those with H-band images. We confirm the detection of a dip in the luminosity function indicative of a discontinuity in the cluster galaxy population; the dip is more pronounced at redder wavelengths. We find, in agreement with earlier works of Tully & Verheijen and ours for Ursa Major cluster galaxies, a clear dichotomy between high and low surface brightness galaxy disks. The difference between the low and high brightness peaks of Virgo disk galaxies is ~2 H-mag arcsec^-2, significantly larger than any systematic errors. The high surface brightness disk galaxies have two distinct classes of high and low concentration bulges, while low surface brightness galaxies have only low concentration bulges. Early-type galaxies exhibit a similar structural bimodality though offset from that of the spiral galaxies towards higher surface brightnesses. Both the early- and late-type structural bimodalities are uncorrelated with colour or any other structural parameter except, possibly, circular velocity. The structural bimodality may be linked to dynamical properties of galaxies. Low angular momentum systems may collapse to form dynamically important disks with high surface brightness, while high angular momentum systems would end up as low surface brightness galaxies dominated by the dark halo at all radii. The confirmation of structural bimodality for gas-rich and gas-poor galaxies in the high-density Virgo cluster as well as the low-density UMa cluster suggests that this phenomenon is independent of environment.