The spectrum of HD 191612 has been found to display large, recurrent variations between two highly reproducible, peculiar states; at least four transformations have occurred since 1950. In one state, the spectral type is O6-O7, with C III λ4650 emission comparable to N III λ4640 (the definition of the Of?p category) and P Cygni profiles at He II λ4686 and Hα. In the other state, the spectral type is O8, with the C III emission absent, very strong N III λ4097 absorption, broad He II λ4686 absorption with narrow central emission (a profile that may be unprecedented in this line among known O-type spectra), and a broad asymmetrical absorption at Hα. One observing sequence over several consecutive nights shows no spectral variations, practically ruling out a short-period, interacting binary as the origin of the phenomenon; moreover, no significant radial velocity variations have been found. Although the sporadic observational record prior to the discovery of the variations in early 2001 precludes definite conclusions, it is possible that a given state is maintained for a decade or longer, but one transformation occurred within 13 months, and the data obtained during 2002 suggest an event with a shorter timescale. The Of?p category currently contains only five members: three in the Galaxy and two in the Small Magellanic Cloud. The other two Galactic members also display bizarre and unexplained phenomena; in the case of HD 108, they are strikingly similar to those described here. Because of their relatively high X-ray luminosities, all three Galactic objects have been suggested to have collapsed companions. If the spectral variations of HD 108 and HD 191612 are due to binary interactions, they are likely multiyear, eccentric systems like WR 140 and η Carinae. The axisymmetric shell ejections of HD 148937 could have a similar origin. Alternatively, these stars may be rapid rotators or in an unstable evolutionary transitional stage. Further intensive spectroscopic monitoring is required to reveal their nature.