A few decades ago, the significance of Moffatt vortices was demonstrated by establishing their existence in various flows. Wedge and cusp regions and their axisymmetric counterparts were preferred to conical regions because the associated analyses were simpler. The lowest even and odd modes were always dominant and the streamline patterns of higher modes were assumed to be similarly simple, especially as their minute strength caused computational difficulties. Here, armed with far more computer power, we return to the vortices' canonical structure, with our principal focus on the region exterior to two cones with common axis and vertex. Many interesting features are revealed, the most unexpected being the structure of the third (second odd in a symmetric geometry) mode. The two-cone geometry allows consideration of asymmetric regions, for the first time. Comparisons are made with the well-known wedge and single-cone results and numerical corrections made to the latter. In all cases, eigenvalue plots play a valuable role in guiding the discussion.