Although most animals are bilaterally symmetric, there are numerous subtle differences between left and right. An obvious example is that most humans prefer one hand, usually the right, over the other. Another famous example is the lateralization of most language functions to the left hemisphere of the brain.
Our perception of the world is not perfectly symmetrical either. In general we pay more attention to what happens to our right, but this preference is heavily dependent on context. When it comes to social interactions, we prefer the left visual field. For example, we tend to hold babies so that the baby faces us from the left.
Beluga mother and calf (Source: Wikimedia Commons)
This left-side preference in social interactions is not limited to humans. Chicks, for example, prefer to view other familiar chicks with their left eye (they also peck more at other chicks if they approach from the left - a social interaction, but not a very friendly one). And in a recent paper in PLoS One, Karenina and colleagues show that baby Beluga whales prefer to swim on their mothers right side, presumably so the baby can view its mother with its left eye. You might wonder if it's not the other way around, so that the mothers prefer to view the babies with their right eye, but apparently this is not the case. The mother whales are pretty much just floating around, and the babies control most of the interactions. Call me soft, but I think this is a pretty cute illustration of lateralization in social interactions.
Karenina, K., Giljov, A., Baranov, V., Osipova, L., Krasnova, V., & Malashichev, Y. (2010). Visual laterality of calf-mother interactions in wild whales. PLoS ONE, 5(11), e13787. doi:10.1371/journal.pone.0013787
Vallortigara, G. (2000). Comparative neuropsychology of the dual brain: A stroll through animals' left and right perceptual worlds. Brain and Language, 73(2), 189-219. doi:10.1006/brln.2000.2303