Asymmetry of the Kolmogorov complexity of online predicting odd and even bits

Symmetry of information states that $C(x) + C(y|x) = C(x,y) + O(\log C(x))$. We show that a similar relation for online Kolmogorov complexity does not hold. Let the even (online Kolmogorov) complexity of an n-bitstring $x1x2... xn$ be the length of a shortest program that computes $x2$ on input $x1$, computes $x4$ on input $x1x2x3$, etc; and similar for odd complexity. We show that for all n there exist an n-bit x such that both odd and even complexity are almost as large as the Kolmogorov complexity of the whole string. Moreover, flipping odd and even bits to obtain a sequence $x2x1x4x_3\ldots$, decreases the sum of odd and even complexity to $C(x)$.