A stationary point of inflection implies a second derivative of 0, does this work the other way around?

No, and we may take a counterexample to see why. If y=x^5+5/3x^4, dy/dx=5x^4+20/3x^3, d2y/dx2=20x^3+20x*2=20x^2(x+1). Setting this to 0 will give us the candidates for a POI, but not all these numbers will be. If we set this to 0 we get 20x^2(x+1)=0 so x=-1, or x=0. But we see that at x=0 on the graph that the stationary point is a minimum.So just having a second derivative of 0 is not sufficient to determine if a point is an inflection, but setting the second derivative to 0 gives all the possible candidates for inflection points.