For a subfactor N ⊂ M with finite Jones index, the intermediate subfactors N ⊂ P ⊂ M correspond to biprojections in the associated planar algebra- 2-boxes which are projections with respect to either shading of the multiplication tangle (Bisch). Bisch and Jones constructed the planar algebra generated by a biprojection, the Fuss-Catalan algebra, which gives a generic (free) composition of two Temperley-Lieb algebras. The situation with two non-commuting biprojections is more complicated, and other invariants such as angles (Sano-Watatani) come into play. We will discuss some recent results on classifying pairs of non-commuting intermediate subfactors. This is joint work with Vaughan Jones and with Masaki Izumi.