Why is the language of even-length non-palindromes context-free? We know $L_1=\{w_1 w_2 \in (a+b)^*\mid |w_1|=|w_2|, w_2 \neq w_1^{\;\mathrm{R}}\}$
is a context-free language.

Can anyone help me produce a PDA or give me any hint how I can quickly understand why this is context-free?
The language of even-length non-palindromes is given by the following context-free grammar:

$$S \rightarrow 0S0 \mid 1S1 \mid D$$
$$D \rightarrow 1A0 \mid 0A1$$
$$A \rightarrow \lambda \mid 00A \mid  01A \mid  10A \mid  11A$$