I'd quibble with some of Rimmer's definitions, but his point is essentially correct.
I believe the real definition of solid state is something more like "made with transistors and/or integrated circuits". An old vacuum tube memory unit has no moving parts, either, but wouldn't be called solid state. Core memory had no moving parts, etc. (I don't mean to be critical; I'm sure someone could quibble with my definitions too. Just trying to clarify.)
Hard disk refers to a technology where data is stored on spinning, rigid disks. We call it hard to distinguish it from floppy disk, where the disks are flexible, and disk to distinguish it from drum storage, where data was stored on the outside of a cylindrical device, i.e. a device that was drum-shaped. (Drum drives have been obsolete for a long time and floppy disks are just about gone, I think.)
A CD is technically a disk that is hard, but we don't call it a hard disk because the technology is quite different and we need to distinguish.
Yes, by definition a hard disk drive is not solid state, just like by definition a cathode-ray tube is not a liquid crystal display, etc. I suppose you could make a disk-shaped solid state memory device, but we wouldn't call it a hard disk drive except as a joke.
Personally I think calling a solid state memory unit a drive is a misnomer, as I understand the word drive to refer to a device with a motor that spins something. But solid-state drives have picked up that name by analogy, I think; they're like a hard disk drive except that they're solid state.
And to ditto Rimmer, they're not really opposites; they are two of many existing and many more possible technologies. I've mentioned seven in this post: core memory, vacuum tubes, drums, hard disks, floppy disks, solid state, and CD. I'm sure in the future we'll see other technologies.