We present observational evidence relevant to the existence of binaries of the morphological type proposed by Wilson (1979), called double-contact binaries. Consider normal evolution with Roche lobe overflow, in the rapid phase of mass exchange. It is well known that such mass-exchange spins-up the accreting star (most of whose mass is transferred material by the end of the rapid phase). Many examples are known (Table 1, for example) of Algol-types with fast-spinning primaries (original secondaries). Suppose the accreting star has been spun-up so much that it cannot accept further high-angular momentum material. It will then be filling its limiting rotational lobe (but not its Roche lobe). The secondary (original primary) will still be filling its Roche lobe. The overall result is that both stars fill their limiting lobes accurately, but are not in contact with one another. Since it is common to speak of “the contact component” of a semi-detached system - meaning the one in contact with its limiting lobe - it seems natural to call members of the new morphological type “double-contact” systems. Of course, there must be a repository for the mass lost by the Roche lobe filling and rotational lobe filling components. This would naturally be the circumstellar disks around the primary components of such systems as β Lyr and V356 Sgr (Wilson and Caldwell, 1978). The end of the double-contact phase would occur when the rate of transfer of rotational angular momentum back into the orbit, by tidal braking, begins to exceed the rate of transfer of angular momentum the opposite way by mass exchange.