is the intersection of a context free language and a regular language a two way street?
Your clam is false even in the special case where $L_1 \cap L_2$ is regular.
To construct a counterexample let $L_1$ be a language that is not context free (e.g.,  $L_1 = \{a^n b^n  c^n: n \ge 0\}$) and pick $L_2 =\emptyset$ (other choices for $L_2$ work too).
Both $L_2 = \emptyset$ and $L_1 \cap L_2 = \emptyset$ are regular (and hence also context-free), but $L_1$ was chosen as a non-context free language.