Firstly, everywhere below I am only considering polynomials of degree $n$ whose derivatives have $(n-1)$ real, distinct roots.
Consider a quadratic polynomial $f(x)=ax^2+bx+c$
It is known that the quadratic polynomial has an axis of symmetry at the root of its derivative, i.e. $x=-\frac{b}{2a}$
Similarly, a cubic polynomial $g(x)=ax^3+bx^2+cx+d$ is centrally symmetric about the root of its second derivative $x=-\frac{b}{3a}$, by which I mean : $$\frac{f(-\frac{b}{3a}+x)+f(-\frac{b}{3a}-x)}{2} = f\left(-\frac{b}{3a}\right)$$
Can we extend this result to all polynomials? Can we say :
All polynomials of even degree $n$ have an axis of symmetry at the root of their $(n-1)th$ dervative, and all polynomials of odd degree $n$ have a centre of symmetry at the root of their $(n-1)th$ dervative