HCN and CO line diagnostics provide new insight into the OH megamaser (OHM) phenomenon, suggesting a dense gas trigger for OHMs. We identify three physical properties that differentiate OHM hosts from other starburst galaxies: (1) OHMs have the highest mean molecular gas densities among starburst galaxies; nearly all OHM hosts have <n(H2)> = 10^3-10^4 cm^-3 (OH line-emitting clouds likely have n(H2) > 10^4 cm^-3). (2) OHM hosts are a distinct population in the nonlinear part of the IR-CO relation. (3) OHM hosts have exceptionally high dense molecular gas fractions, L(HCN)/L(CO)>0.07, and comprise roughly half of this unusual population. OH absorbers and kilomasers generally follow the linear IR-CO relation and are uniformly distributed in dense gas fraction and L(HCN), demonstrating that OHMs are independent of OH abundance. The fraction of non-OHMs with high mean densities and high dense gas fractions constrains beaming to be a minor effect: OHM emission solid angle must exceed 2 pi steradians. Contrary to conventional wisdom, IR luminosity does not dictate OHM formation; both star formation and OHM activity are consequences of tidal density enhancements accompanying galaxy interactions. The OHM fraction in starbursts is likely due to the fraction of mergers experiencing a temporal spike in tidally driven density enhancement. OHMs are thus signposts marking the most intense, compact, and unusual modes of star formation in the local universe. Future high redshift OHM surveys can now be interpreted in a star formation and galaxy evolution context, indicating both the merging rate of galaxies and the burst contribution to star formation. Comment: 5 pages, 3 figures, 1 table, accepted by ApJ Letters