The stress-energy tensor is an actual tensor, i.e., under a spacetime parity transformation it stays the same, which is what a tensor with two indices is supposed to do according to the tensor transformation law. This also makes sense because in the Einstein field equations, the stress-energy tensor is related to the Einstein tensor, which is tensorial. However, in three-dimensional continuum mechanics, the stress tensor takes a normal vector as an input and gives a stress vector as an output. In three dimensions, a normal vector is an axial vector (even under parity), while a stress vector is a true vector (odd under parity). Therefore it seems that the stress 3-tensor must be odd under parity, which makes it not a real tensor. Is this analysis correct? I'm used to thinking of the stress 3-tensor as a block of elements in the stress-energy tensor, when they're expressed in Minkowski coordinates. Doesn't that imply that they should have the same parity properties?