Given the binary operation \(a \times b = a^b\), and \(a \times 2 = 2 - a\), we can substitute the definition of the operation into the equation to get:

\(a^2 = 2 - a\)

This is a quadratic equation, and its solutions can be found using the quadratic formula:

\(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Substituting \(a = 1\), \(b = 1\), and \(c = -2\) into the formula, we get:

\(a = \frac{-1 \pm \sqrt{1^2 - 4*1*(-2)}}{2*1}\)

This simplifies to:

\(a = \frac{-1 \pm \sqrt{1 + 8}}{2}\)

So the solutions are:

\(a = \frac{-1 + \sqrt{9}}{2} = 1\)

and

\(a = \frac{-1 - \sqrt{9}}{2} = -2\)

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