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Analyses that connect astrophysical observations of neutron stars withnuclear matter properties sometimes rely on equation-of-state insensitiverelations. We show that the slope of the binary Love relations (i.e.~betweenthe tidal deformabilities of binary neutron stars) encodes the rate of changeof the nuclear matter speed of sound below three times nuclear saturationdensity. Twin stars lead to relations that present a signature ''hill'',''drop'', and ''swoosh'' due to the second (mass-radius) stable branch,requiring a new description of the binary love relations.

The slope, the hill, the drop, and the swoosh: Learning about the nuclear matter equation of state from the binary Love relations