A symmetric key schedule is a key schedule of the form $(k_1,\dots,k_{n-1},k_n,k_{n-1},\dots,k_1)$ or of the form $(k_1,\dots,k_{n-1},k_n,k_n,k_{n-1},\dots,k_1)$. A skew symmetric key schedule is a key schedule of the form $(k_1,\dots,k_{n-1},k_n,g(k_{n-1}),\dots,g(k_1))$ or of the form $(k_1,\dots,k_{n-1},k_n,g(k_n),g(k_{n-1}),\dots,g(k_1))$ for some bit permutation $g$.
What kinds of attacks are block ciphers with symmetric key schedules susceptible to? For example, if we replace the AES-128 key schedule $(k_0,\dots,k_{10})$ with the symmetric key schedule $(k_0,\dots,k_4,k_5,k_4,\dots,k_0)$, or a skew symmetric key schedule of the form $(k_0,\dots,k_4,k_5,g(k_4),\dots,g(k_0))$, what could go wrong?
If seems like the two instances of the round key $k_0$ are far enough apart from each other that reusing this round key may not be disastrous, but the two instances of $k_{n-1}$ are much closer together.
Motivation
If one were to design a block cipher to be efficient for reversible computation, then ideally reversible computation should not produce any computational overhead. This means that to encrypt or decrypt, any uncomputation should be built into the block cipher in order to provide extra confusion and diffusion. It seems like having a symmetric key schedule may be a good balance between efficiency and security under reversible computing.
Another option for constructing a reversible key schedule would be to have a cyclic key schedule $(k_0,\dots,k_n)$ where we always have $k_n=k_0$, but having $k_n=k_0$ without needing to uncompute the round keys does not leave us with too many options.
