How is the Lai-Massey scheme invertible? The Wikipedia page on the subject and other descriptions that I can find of it are as clear as mud. $L$ and $R$ are combined to make a number that is then added to both. That seems like a loss of information to me.
One Lai-Massey round can be described as 
$$
\begin{align}
L' &amp;= \sigma(L \oplus F_k(L \oplus R)) \\
R' &amp;= R \oplus F_k(L \oplus R)\,,
\end{align}
$$
where $F_k$ is some round function—not necessarily invertible—and $\sigma(\cdot)$ is an orthomorphism, an arbitrary function such that both $\sigma(x)$ and $\sigma'(x) = \sigma(x) \oplus x$ are invertible.

To invert this,
$$
\begin{align}
L &amp;= \sigma^{-1}(L') \oplus F_k(\sigma^{-1}(L') \oplus R') \\
R &amp;= R' \oplus F_k(\sigma^{-1}(L') \oplus R')\,.
\end{align}
$$
We can see this works because
$$
\begin{align}
\sigma^{-1}(L') &amp;= 
L \oplus F_k(L \oplus R)\,,
\end{align}
$$
and
$$
\begin{align}
\sigma^{-1}(L') \oplus R' &amp;= 
L \oplus F_k(L \oplus R) \oplus R \oplus F_k(L \oplus R) \\
&amp;= L \oplus R\,.
\end{align}
$$