We show that
K6(2)∩X6(2)=D6(1)∪C6(1) up to standard Hadamard equivalence, where
C6(1) is a new non-affine one-parameter family of complex Hadamard matrices. This demonstrates that the intersection of the two-parameter Hadamard families of Karlsson and Szöllősi is strictly larger than the known affine Diţă curve
D6(1). Our new parametrisation of the Szöllősi family
X6(2) in Karlsson coordinates and our constructive classification of the Karlsson-Szöllősi intersection shed light on the intersection of block-reducible forms and hypocycloid constraints. This may support the search for mutually unbiased bases in six dimensions.