| A search for maser emission at the 1720-MHz transition of OH has been made with the Australia Telescope Compact Array (ATCA) towards more than 200 southern sites of 1665- and 6035-MHz OH masers. 28 of the 36 detected 1720-MHz masers precisely coincide with the targets (15 previously known; 13 new discoveries) and are thus almost certainly of the star formation region (SFR) variety. Our observations revealed highly variable intensities for some previously known 1720-MHz masers, in extreme cases now preventing their detection. We interpret six new masers serendipitously discovered offset from any target position as an SFR subclass. Two other masers were chance detections of a quite different maser class associated with supernova remnants. Several instances of prominent 1720-MHz OH absorption occur. The 1720-MHz masers mostly coincide, to arcsecond accuracy, with a stronger 1665-MHz OH maser counterpart; in a few instances, 1665-MHz counterparts are much weaker, and the six 1720-MHz masers discovered offset from any target position (with no detectable 1665-MHz emission) are the ultimate examples, qualifying them as a distinct SFR subclass. At the sensitivity of the current survey, we find that SFR 1720-MHz masers are approximately one-sixth as common as 1665-MHz masers. The 1720-MHz masers preferentially occur at the sites hosting 6035-MHz masers - explicable in the context of current maser pumping models if 1720-MHz and 6035-MHz masers trace regions where the OH column density is higher, and the kinetic temperature lower, than at sites displaying only 1665-MHz masers. For each 1720-MHz maser, spectra of the two circular polarizations at high-velocity resolution were taken with the Parkes telescope. Similar observations, to monitor several previously known northern 1720-MHz masers, augment the southern sample. The Parkes spectra commonly reveal Zeeman splitting in magnetic fields of a few milligauss, and the largest field is a `record-setting' 16 mG. Compared with fields derived from other OH transitions at the same site, the 1720-MHz values generally imply the same direction of field, but the field strength tends to be higher, by as much as a factor of 2. This may be understood if the 1720-MHz masers preferentially occur at locations compressed to higher density, with enhanced frozen-in magnetic field. Such higher fields support a pumping scheme requiring high densities for the 1720-MHz masers. Towards most of the 1720-MHz maser sites lies an associated 6668-MHz methanol maser, consistent with pumping expectations that the methanol masers are also favoured by high densities. We also explore some sites in more detail, with new results obtained in the radio continuum and recombination lines, and at other masing transitions - a new position for 328.809+0.633 at the 1612-MHz OH transition, for example, confirms a rare spatial near-coincidence with a 1720-MHz maser (although offset in velocity). |