Heat and mass integration to consolidate distillation columns in a configuration is often characterized by elimination of a reboiler and condenser associated with the same components. Here, we study a new and more general approach to column consolidation, of which the conventional approach is a special case. In the new approach, the reboiler of a column is coupled with the condenser of another, through heat and mass integration in an additional section (HMA-section). The introduction of an HMA-section eliminates multiple connecting streams/valves, reduces the number of reboilers, condensers, and distillation columns by one each, and reduces the heat duty of the configuration. We exhaustively enumerate HMA-sections, and lay out a framework to identify all configurations with HMA-sections. Through examples, we show that both heat integration and mass integration resulting from introducing an HMA-section contribute to reduce the heat duty significantly.