Two kinds of indices, covariant and contravariant, are introduced in special relativity. This, as far as I understand, is solely for mathematical luxury, i.e. write expressions in a concise, self-explanatory notation. For example, instead of writing the metric as $(\Delta s)^2=c^2(\Delta t)^2-(\Delta \textbf{r})^2$ one can write $x^\mu x_\mu$ which is not only a compact notation but also tells us that this expression is Lorentz invariant. But both $x_\mu$ and $x^\mu$, represent same objects: a set of four co-ordinates $(ct,x,y,z)$.

In the case of representations of $\mathrm{SU}(N)$, there too appear objects such as $\psi^i$ and $\psi_i$ which transform differently but keep $\psi_i\psi^i$ invariant. But we see that two different kind of objects exist in nature: quarks and anti-quarks which belong to the representations $\psi^i$ and $\psi_i$ respectively.

Does it mean in the latter case the distinction between covariant $\psi_i$ and contravariant $\psi^i$ is more fundamental than in the former case?