The orbits of giant extrasolar planets often have surprisingly small semimajor axes, large eccentricities, or severe misalignments between their orbit normals and their host stars' spin axes. In some formation scenarios invoking Kozai-Lidov oscillations, an external planetary companion drives a planet onto an orbit having these properties. The mutual inclinations for Kozai-Lidov oscillations can be large and have not been confirmed observationally. Here we present evidence that observed eccentric warm Jupiters with eccentric giant companions have mutual inclinations that oscillate between 35° and 65°. Our inference is based on the pairs' observed apsidal separations, which cluster near 90°. The near-orthogonality of periapse directions is effected by the outer companion's quadrupolar and octupolar potentials. These systems may be undergoing a stalled version of tidal migration that produces warm Jupiters over hot Jupiters, and they provide evidence for a population of multiplanet systems that are not flat and have been sculpted by Kozai-Lidov oscillations.
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