Double Cryptogram

Logic Level 5

$\large{\begin{array}{cccccc} & & & a & b & c&d\\ +& & & b & c & d & a \\ \hline && & & & & \phantom0&\\ \end{array}} \qquad \qquad \large{\begin{array}{cccccc} & & & d & c & b &a \\ +& & & c& b & a & d \\ \hline && & & & \phantom0 &\\ \end{array}}$

If $a,b,c$ and $d$ are distinct single-digit positive integers so that the result of both additions is the same, find the number of possible quadruples $(a,b,c,d)$ .

See a harder version of this problem.