Given that \(a * 2 = 2 - a\), let's substitute \(2\) for \(b\) in the definition of the binary operation:

\[a * 2 = a^2\]

Now we have the equation: \(a^2 = 2 - a\).

Rearrange the equation to a quadratic form:

\[a^2 + a - 2 = 0\]

Now, let's factor the quadratic equation:

\[(a + 2)(a - 1) = 0\]

Setting each factor to zero and solving for \(a\):

\(a + 2 = 0 \implies a = -2\)

\(a - 1 = 0 \implies a = 1\)

So, the possible values of \(a\) are \(1\) and \(-2\).