$Assumptions = {
   Element[t1, Arrays[{3, 3, 3}, Reals, Antisymmetric[{1, 2, 3}]]]
   , Element[t2, Arrays[{3, 3}, Reals, Symmetric[{1, 2}]]]
   };
t = TensorProduct[t1, t2, t2];
ts = TensorSymmetry@t

p = Permutations@Range@TensorRank@t;

pli = Select[p,
   Not[TensorReduce@TensorTranspose[t, #] === t ||
      TensorReduce@TensorTranspose[t, #] === -t] &];

Length@p
Length@pli