Symmetry in exponential sums

Today's exponential sum is highly symmetric:

These sums are often symmetric, but not always. For example, here's the sum from a couple days ago:

It's not obvious from looking at the parameters whether a sum will be symmetric or not. Maybe someone could find a prove criteria for a sum to have certain symmetries.

For more exponential sums, see this page. There's a link labeled details that explains what is being plotted, and each page has an animate link to show the steps in creating that image.