According to Wikipedia and validated by a clever thought experiment here, sound waves can be transverse as well as longitudinal, if they're propagating through a solid. Consider my mind blown and my curiosity piqued. However, is this a phenomenon we can hear? And is there any reason these waves would be different from longitudinal sound waves?
Understandably, this kind of sound wouldn't travel through the normal path as that requires traveling through the air to get to the eardrum. Bone conduction headphones seem to offer a promising avenue, but I see no reason that those sound waves wouldn't be longitudinal.