Symmetry is part of beauty across time and culture. For example people with more facial symmetry are considered better looking.

Yes, but it's not evidence of intent. It's lack of deformity and disease.

The word for "lack of deformity" is "form" which we generally use to refer to something which was generated based on a rule, as opposed to complete randomness.

Which is evidence of intent! -- the intent of the genome being successfully carried out. :-)

Perhaps? Can you think of a counter-example, where seeing an event happen once is more attention-getting and memorable than seeing the event happen twice in a row in quick succession?

Imagine you see a herd of deer trotting along, and one goes down. A hole, or ditch, or maybe a predator? Now imagine two deer go down at nearly the same time. Different implications, or at least different probabilities to the possible implications.

I'll stand by my over-generalization for now, but I'm open to counter-examples.