There's no mode bit in the hardware.
Yes there is (MSR[SF]). It mainly masks the upper 32 bits of the effective address, but also affects certain overflow and carry calculations.
The 32-bit ISA is just a subset of the 64-bit. Code that restricts itself to the 32-bit ISA can access only the low 32 bits of the integer registers.
That is correct, in that the 32-bit subset cannot "see" the upper bits, but it can change them. And, of course, a process in 32-bit mode can use the full 64-bit instruction set.
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