The decays of χc2→K+K-π0, KSK±π, and π+π-π0 are studied with the ψ(3686) data samples collected with the Beijing Spectrometer (BESIII). For the first time, the branching fractions of χc2→K∗K̄, χc2→a2±(1320)π/a20(1320)π0, and χc2→ρ(770)±π are measured. Here, K∗K̄ denotes both K∗±K and its isospin-conjugated process K∗0K̄0+c.c., and K∗ denotes the resonances K∗(892), K2∗(1430), and K3∗(1780). The observations indicate a strong violation of the helicity selection rule in χc2 decays into vector and pseudoscalar meson pairs. The measured branching fractions of χc2→K∗(892)K̄ are more than ten times larger than the upper limit of χc2→ρ(770)±π, which is so far the first direct observation of a significant U-spin symmetry breaking effect in charmonium decays.