Interpretation of the Cross Partials of the Cobb-Douglas Consider a Cobb-Douglas Prod. Function $$Y=AL^{a}K^{1-\alpha}$$
This has the cross-partial:
$$\frac{\partial^2 Y}{\partial K\partial L}=(1-\alpha)\alpha AL^{\alpha-1}K^{-\alpha}$$

Is the interpretation of this the change in the return to labour, with a change in capital (and by Schwarz's theorem, also the change in the return the capital, with a change in labour)?

Then, is it correct to say that the return to labour increases as $\frac{1}{K^a}$ with capital?
The interpretation of the cross partial here is the change in the return (or marginal productivity) of labor when capital increases marginally. Similarly, it can be interpreted as the change in the return of capital, when labor increases marginally.

We often only want to interpret the sign of the cross-partial and not the exact value. Here we see that the productivity of one factor increases with the other factor. Interpreting the exact value is more difficult, but also not really interesting for most economic applications.

Furthermore, it is not correct to say that the return to labour increases as $\frac{1}{K^a}$ with capital. How did you arrive at this conclusion?