Lenses are programs that can be run both "front to back" and "back to front," allowing updates to either their source or their target data to be transferred in both directions. Lenses have been extensively studied, extended, and applied. Recent work has demonstrated how techniques from type-directed program synthesis can be used to efficiently synthesize a simple class of lenses---bijective lenses over string data---given a pair of types (regular expressions) and examples. We extend this synthesis algorithm to a broader class of lenses, called simple symmetric lenses, including all bijective lenses, all of the popular category of "asymmetric" lenses, and a subset of the "symmetric lenses" proposed by Hofmann et al. Intuitively, simple symmetric lenses allow some information to be present on one side but not the other and vice versa. They are of independent theoretical interest, being the largest class of symmetric lenses that do not use persistent internal state. Synthesizing simple symmetric lenses is more challenging than synthesizing bijective lenses: Since some of the information on each side can be "disconnected" from the other side, there will typically be many lenses that agree with a given example. To guide the search process, we use stochastic regular expressions and information theory to estimate the amount of information propagated by a candidate lens, preferring lenses that propagate more information, as well as user annotations marking parts of the source and target formats as either irrelevant or essential. We describe an implementation of simple symmetric lenses and our synthesis procedure as extensions to the Boomerang language. We evaluate its performance on 48 benchmark examples drawn from Flash Fill, Augeas, and the bidirectional programming literature. Our implementation can synthesize each of these lenses in under 30 seconds.