Given formula for a linear interpolation, I can solve for $x$ as follows: \begin{align} y = a(1-x)+bx\\ y = a+x(b-a)\\ x = \frac{y-a}{b-a} \end{align}
How do I solve for $x$ given formula for cubic interpolation: $$ y = a(1-x)^3 + 3bx(1-x)^2 + 3cx^2(1-x) + dx^3 $$
You can find an analytical formula for $x$ here: https://en.wikipedia.org/wiki/Cubic_equation#Trigonometric_and_hyperbolic_solutions (however first you have to transform your cubic equation to the standard form $x^3+Bx^2+Cx+D=0$ and further continue by substitution $y=x-\frac{B}{3}$ to get the so-called depressed form $x^3+Ex+F=0$).
Or you can find $x$ numerically, by means of the Newton's method for example, which however gives you only one of the solutions.