Take a non-degenerate polygon with side lengths $\{a_1,\dots,a_n\}$ in a convex configuration. What is the condition on the $a_i$'s so that the polygon can be turned inside out by a continuous motion that preserves the side lengths.

For example any triangle cannot be, but the unit square and unit hexagon can be, since you can collapse both to form a degenerate shape with zero area. A four sided figure can sometimes be inverted, sometimes not if the smallest side is too small. 

Just working out the four sided figure could be interesting I think.

I suspect that it is possible iff you can find a configuration of zero area, where the area is defined as $\frac{1}{n}(x_1y_2-x_2y_1+x_2y_3-x_3y_2+\dots+x_ny_1-x_1y_n)$ where the $(x_i,y_i)$ are vertices of the polygon with i increasing from 1 to n as you progress around the polygon in order. It is trivial that if a deformation exists the area must change sign and hence be zero but the reverse implication may be more challenging. (Of course this doesn't solve the problem as this is not a condition on the side lengths but might be a useful observation).