Luckily, Eric soon discovered a few others in the water:
We could tell it was a jellyfish, with noticeable reddish-brown stripes made up of leopard-like spots. However, I was having trouble identifying the species at first. (I suppose this shows that I've been away from the Cape for too long!)
Eventually we found one in a little better shape, showing some of the frilly underparts and a few tentacles:
When we reviewed possibilities in a field guide, we still weren't totally certain about its identification. But after searching the Internet for photographs, and comparing it with similar species, we feel good about these being Atlantic Sea Nettles (Chrysaora quinquecirrha).
Apparently, they tend to have darker patterning like this when living in high salinity water; they tend to be paler (and plainer) when living in more estuarine conditions.
Here's one more picture that I took with one of the jellies in a plastic bag filled with water. You can see the leopard-like spots and the granular texture on the bell.
Atlantic Sea Nettles are generally a more southern species. Cape Cod is their northern range limit.
P.S. Note that Atlantic Sea Nettles are known for painful stings, so if you see one, be careful not to handle it directly.