Hi,
Joy of Cooking has an interesting section on substitutions. Mayonnaise is a substitute for eggs. Given the close proximity to shops in Metropolitan areas, this is an interesting detail. However, for those out in the country, it is very useful information when the shop is 20 miles away.
Typical boxed cake instructions have you including eggs, oil and water. Mayonnaise itself is an emulsion of eggs and oil, so it is not as far fetched as it sounds.
A long time ago, I used mayonnaise as a substitute (just for chuckles). I did notice a texture difference. I cannot recall if there was a big shift in the taste. I do recall thinking I would only do this again if I was in a pinch.
I hope you will try it and update our information.