Vector Spaces & Bases · Challenge problem

Let $V$ be the set of real polynomials $p$ of degree at most $6$ satisfying $x^6\,p(1/x) = p(x)$ (the "palindromic" polynomials, such as $1 + 2x + 5x^3 + 2x^5 + x^6$). What is $\dim V$?