x86 processors operate in one of several modes. 64-bit "long" mode is an additional mode introduced on top of the plethora of modes already present among 32-bit CPUs. 64-bit CPUs still retain all the old modes, just as 32-bit CPUs retain the older 16-bit mode of the old 8086.
Oversimplifying just a bit, as the x86 architecture has progressed through the years since the original 8086 CPU from the late 70's (itself a derivative of an even older CPU, the 8080), additional modes have been tacked on to it.
The 80386 supported something called 32-bit or "protected" mode. However, it boots up in the original 16-bit 8086 mode, for compatibility. One of the tasks of an operating system initializer is to switch it to this 32-bit protected mode, on a 32-bit CPU.
Beginning with AMD's 64-bit extensions to this (Intel for a short time had a competing standard), there is yet another mode tacked on called 64-bit "long mode". However, at least on BIOS-based systems (UEFI might be different), such CPUs still boot up in that old 16-bit mode. When the OS gets everything set up, it needs to switch the CPU into 64-bit mode to run 64-bit code.
So every 64-bit CPU is also a 32-bit CPU, and a 16-bit CPU as well. Just depends on what mode you put the CPU in.