In venue of my old question When polynomial f(x^2) can be factored as g(x)·g(-x)? and this recent answer to a different question, I wonder:

How to characterize polynomials $f(x)$ with rational coefficients such that $f(t+t^{-1})=g(t)\cdot g(t^{-1})$, where $g(x)$ is also a polynomial with rational coefficients?

Is there a computationally efficient way to identify if a given polynomial $f(x)$ is such, without factoring $f(t+t^{-1})$ ?