Is an intersection between NPC and coNPC always DPC?
NO, it is not always so.

$L_1=\{&lt;0,\varphi&gt;|\varphi\ \mathrm{is\ satisfiable}\}$

$L_2=\{&lt;1,\varphi&gt;|\varphi\ \mathrm{is\ unsatisfiable}\}$

$L_1\cap L_2=\emptyset$

EDIT: As commented below by David Richerby, $L_1=SAT$ and $L_2=UNSAT$ would likewise falsify your claim in the question.