Property of odd degree polynomials?
Partial answer: For the case of cubic polynomials it's true.
Let $ax^3+bx^2+cx+d$ be a cubic polynomial in $x$ with all coefficients positive and three real zeroes. All three must be negative and they sum to $-b/a$, thus all are greater than $-b/a$. This forces the polynomial value at $x=-b/a$ to be negative, but that value is $(ad-bc)\over a$.
Thus $ad<bc$, but that can't be true for all permutations of a given set of positive numbers. So at least one permutation of the coefficients fails to give three real roots.