Orthogonally polarized beams do not interfere, do they beat?

I'm trying to understand what happens with beating light when the componenets have differing polarisation.

I know that orthogonally polarised beams do not interfere spatially. I looked in Hecht and this arises in the maths due to the 'interference term' in which there is a $E_{1}.E_{2}$ term. Obviously if they are orthogonal the dot product is zero.

However when I look up the treatment for beating (which is before) the electric fields are just treated as scalars, so the polarisation is not considered. As the above, vector treatment, is more 'complete' I'm inclined to believe that one, however I'm not sure it's correct.

The reason I'm not sure is that in previous thinking about this problem I decided that two co-linear beams of slightly different frequency (as in the beating example) is equivalent to a single frequency beam of rotating polarisation, if you consider that then the changing polarisation will still produce intensity modulation. Hence I'm confused!