Why these cross-connections in the allpass filter?
This is called a lattice structure implementation of all-pass filters.

$k_1$ and $-k_1$ are scalar gains. The input to the delay element is $$a[n]=x[n]-k_1a[n-1]$$ taking the $z$-transform, the transfer function is 
$$A(z)(1+k_1z^{-1})=X(z)\Rightarrow A(z)=\frac{X(z)}{1+k_1z^{-1}}$$ and the output is
$$\begin{align}
y[n]&amp;=k_1a[n]+a[n-1]\\
\end{align}$$
Thus
$$\begin{align}
Y(z)&amp;=A(z)(k_1+z^{-1})\\
&amp;=X(z)\frac{k_1+z^{-1}}{1+k_1z^{-1}}
\end{align}$$
$$H(z)=\frac{Y(z)}{X(z)}=\frac{k_1+z^{-1}}{1+k_1z^{-1}}$$
So it is an all-pass filter ($|H(e^{j\omega})|=1,\ \forall \omega$) with mirrored coefficients in numerator and denomerator.