Studies of solar-type binaries have found coplanarity between the equatorial and orbital planes of systems with < 40 AU separation. By comparison, the alignment of the equatorial and orbital axes in the substellar regime, and the associated implications for formation theory, are relatively poorly constrained. Here we present the discovery of the rotation period of 3.32 +/- 0.15 h from 2MASS J0746+20A - the primary component of a tight (2.7 AU) ultracool dwarf binary system (L0 + L1.5). The newly discovered period, together with the established period via radio observations of the other component, and the well constrained orbital parameters and rotational velocity measurements, allow us to infer alignment of the equatorial planes of both components with the orbital plane of the system to within 10 degrees. This result suggests that solar- type binary formation mechanisms may extend down into the brown dwarf mass range, and we consider a number of formation theories that may be applicable in this case. This is the first such observational result in the very low mass binary regime. In addition, the detected period of 3.32 +/- 0.15 h implies that the reported radio period of 2.07 +/- 0.002 h is associated with the secondary star, not the primary, as was previously claimed. This in turn refutes the claimed radius of 0.78 +/- 0.1 R-J for 2MASS J0746+20A, which we demonstrate to be 0.99 +/- 0.03 R-J.