Symmetry and skew symmetry of homogeneous components of Baker-Campbell-Hausdorff formula

Statement
Consider the Baker-Campbell-Hausdorff formula in two variables $$X,Y$$. For the case of a simply connected group, or a nilpotent group, there is an explicit expression of the formula as a $$\mathbb{Q}$$-linear combination of $$X,Y$$ and their brackets. For any positive integer $$d$$, denote by $$t_d(X,Y)$$ the degree $$d$$ homogeneous component of this expression. Then, we have that:

$$t_d(X,Y) = (-1)^{d-1}t_d(Y,X)$$

In other words, the degree $$d$$ homogeneous component is symmetric in $$X,Y$$ if $$d$$ is odd and is skew symmetric in $$X,Y$$ if $$d$$ is even.

Note that, except in the case $$d = 2$$, the degree $$d$$ homogeneous component is not bilinear, hence these terms symmetric and skew symmetric are meant without the bilinear connotation.