As answered in this question, an antisymmetric tensor on 4D Minkowski space has two Lorentz-invariant degrees of freedom. These are the two scalar combinations of the electromagnetic tensor (as proven for example in Landau & Lifshitz' "Classical Theory of Fields"):
$$ L_1 = \eta_{\mu\rho} \eta_{\nu\sigma} F^{\mu\nu} F^{\rho \sigma} \,, \qquad L_2 = \varepsilon_{\mu\nu\rho\sigma} F^{\mu\nu} F^{\rho \sigma} \,, $$
and any other Lorentz-invariant monomial in the components of $F^{\mu\nu}$ can be written as a function of $L_1$ and $L_2$.
My question is, is there a way to establish the number of Lorentz-invariant degrees of freedom of an arbitrary tensor? The solution in Landau-Lifshitz for antisymmetric tensors involves writing the complexified version of $F$, and Lorentz transformations as elements of $SO(3,\mathbb{C})$, and then using the fact that the solution of the same problem for complex vectors is already known.
I know that one can enumerate all Lorentz-invariant products of $m$ copies of a Lorentz n-tensor $T^{\mu_1 \dots \mu_n}$ by finding all possible partitions of $m \times n$ indices into groups of two and four, and then saturating $T^{\mu_1 \dots \mu_n}T^{\mu_{n+1} \dots \mu_{2n}} \dots T^{\mu_{(m-1) \cdot n +1} \dots \mu_{m\cdot n}}$ with products of $\varepsilon_{\mu\nu\rho\sigma} $ and $\eta_{\mu\nu}$ tensors in all possible ways. However, this gives an infinite number of homogeneous polynomials in the components of $T^{\mu_1 \dots \mu_n}$, and all of them must be expressible as functions of a finite number $N_{dofs}$ of them. I would like to know if, given n and given the symmetries of $T^{\mu_1 \dots \mu_n}$, there is a way to determine $N_{dofs}$ and, possibly, to identify $N_{dofs}$ independent scalars such that all other scalars can be written as functions of them.
