# Complex pseudotensor generalisation [closed]

A tensor $$T$$ is an object which is invariant under all coordinate transformations: $$T\mapsto T = e^{i0}T.$$ A pseudotensor $$P$$ is an object which changes its sign under the inversion of a coordinate axis: $$P\mapsto P = e^{i\pi}P.$$ As shown above, these are two special cases of the more general mapping $$G \mapsto e^{i\theta}G$$ for some generalised object $$G$$ under the inversion of a coordinate axis. Do any objects exist which have $$\theta\neq 0, \pi$$? If so, where do they arise and what are they called?

For example, if you switch from XYZ-basis to circular basis, i.e. +-0, then rotation by angle $$\theta$$ around z-axis will act on a tensor $$V^{\alpha_1\dots\alpha_n}_{\beta_1\dots\beta_n}$$ as:
$$V^{\alpha_1\dots\alpha_n}_{\beta_1\dots\beta_n} \overset{R_z\left(\theta\right)}{\to} \exp\left(i\theta\cdot\left(\alpha_1+\alpha_2+\dots\alpha_n-\beta_1-\beta_2-\dots\beta_n\right)\right)V^{\alpha_1\dots\alpha_n}_{\beta_1\dots\beta_n}$$
Where $$\alpha_k=\pm 1,0$$ etc. But this is coordinate-specific.