Let $P(x)$ be an irreducible monic polynomial of degree $\ge4$ with integer coefficients. We all know that over a finite field $\mathbb F_p$, $P$ will often split, and I am interested in polynomials with the property that, for any given $p$, **all splitting factors in $\mathbb F_p[x]$ have the same degree**. Call such polynomials **"uniformly splitting"**. (This doesn't exclude the possibility that for certain $p$, they may remain irreducible.)

E.g. it is not hard to see that $P(x)=x^4-x^2+1$ is uniformly splitting, because it will never split into two linear factors and one quadratic irreducible one.

I suspect the situation is similar for $P(x)=x^8-x^6+x^4-x^2+1$. (Is it really?)

On the other hand, e.g. $P(x)=x^8-x^6-x^4-x^2+1$ splits uniformly for all $p<43$, so far so good, but in $\mathbb F_{43}$, we have
$$P(x)=(9+x) (19+x) (24+x) (34+x) (14+x^2) (40+x^2).$$

What are necessary or sufficient conditions for an irreducible polynomial to be uniformly splitting?

It seems to me like such polynomials are very rare, at least for degree $>4$. I conjecture that cyclotomic polynomials $\Phi_n(x^k)$, as long as they are irreducible, are uniformly splitting.

Given that the constant term must be $\pm1$, it seems reasonable to expect most if not all uniformly splitting polynomials to be symmetric (in the sense of having self-mirrored coefficients). Then all factors of a (uniform) split for a given $p$ look somewhat like "conjugates" of each other, in a yet-to-define broader way obviously involving the primitive elements of $\mathbb F_{p}$. Might Galois theory be of any help here? And:

Are there uniformly splitting polynomials of degree $>4$ which are

notof the form $\Phi_n(x^k)$?