I think it's fairly likely the Saxons had cavalry. The problem is that these days we draw a firm line between mounted infantry and "proper" cavalry, whilst in the Dark Ages the line was heavily blurred between the two.

You are right. Specialy after the 8 century, we know that elites like the later called Houskarls rode to the Battle but fought on ground. For the earlyer anglosaxons, like her around 620 AD, we have not just iconographic references for mounted warrioers (also fighting from horseback) but also a lot of archeological references, which gives us the image, that they had a small elite of mounted warrioers. They disapeared during the change from pagans to christianity and transformed to warriors like the Housekarls.

References for this are many burials, Sutton Hoo Burial mound 17 is one of a few, comparable burials in the origin countries of the anglo-saxons during the same time, like Rullstorf, Sarsted ,..
Iconography, like the Pressbleche from Sutton Hoo ship burial 1, Staffordshire and similar pieces like the Pliezhausen Fibula.
And finaly we have some skeletons which show damages on the bones, which are very plausbile for cavallery, Example the Warrior of Esch- Alzette, around 650. 195 tall skeleton, with a deep cut to his skull in nearly perfect vertical angle, so very unlikely that he became this deadly wound from a enemy also fightung on foot or when looking at the angle during an execution.

These are just references, we have no absolut perfect proof, not for the anglo-saxon's but they allow us a clearer and more likely image.