There are two formal semi-infinite Laurent series

$$ f_+(z)=z+\sum_{k=2}^{\infty} a_k z^k $$

and

$$ f_-(z)=z+\sum_{k=0}^{\infty} b_k z^{-k} $$

Their composition (we assume that this composition exists) $f_+(f_-(z))$ is a formal series infinite in both directions. Is there any way to construct two other semi-infinite series

$$ g_+(z)=z+\sum_{k=2}^{\infty} c_k z^k $$

and

$$ g_-(z)=z+\sum_{k=0}^{\infty} d_k z^{-k} $$

such that

$$f_+(f_-(z))=g_-(g_+(z))$$

How should I call this problem? Are there any non-trivial examples when it can be done?