Is a relation that is purely reflexive also symmetric?

I've seen a useful method for getting a more intuitive understanding of some of these properties of relations, namely using a table to indicate a relation and then visually deduce the property. In the case you describe one obtains:

For example, if a relation is reflexive, the diagonal elements will all be populated with 1's. Note that a "1" indicates that the corresponding row entry and column entry is in the relation, and a "0" means that combination in not in the relation. For example, $left(a,aright) in R$, but $left(a,bright) notin R$, etc. in the above table.

If the relation is symmetric, then the table will identical if you reflect it about the diagonal. Therefore, for the relation you have described in the question, this is clearly true.

Here is another question in Stack Exchange that illustrates this idea further: Checking the binary relations, symmetric, antisymmetric and etc

I hope this helps.