Let $S = \{a, b, c, d\}$. We want to find the number of relations that are both reflexive and symmetric.

A relation $R$ on $S$ is a subset of $S \times S$. Since $|S| = 4$, $|S \times S| = 4 \times 4 = 16$.

For a relation to be reflexive, it must contain the pairs $(a, a), (b, b), (c, c), (d, d)$.

For a relation to be symmetric, if $(x, y) \in R$, then $(y, x) \in R$.

The pairs $(a, a), (b, b), (c, c), (d, d)$ must be in the relation since it is reflexive. These pairs do not affect the symmetry condition.

The remaining pairs in $S \times S$ are:

$(a, b), (a, c), (a, d), (b, a), (b, c), (b, d), (c, a), (c, b), (c, d), (d, a), (d, b), (d, c)$

Since the relation must be symmetric, we can consider the pairs $\{(a, b), (b, a)\}, \{(a, c), (c, a)\}, \{(a, d), (d, a)\}, \{(b, c), (c, b)\}, \{(b, d), (d, b)\}, \{(c, d), (d, c)\}$.

There are 6 such pairs. For each pair, we can either include both elements in the relation or exclude both elements from the relation. Thus, there are 2 choices for each pair.

Therefore, the number of reflexive and symmetric relations is $2^6 = 64$.