REFIND EFI v.1

First, I'd be absolutely shocked if you could run both refind_x64.efi
and refind_ia32.efi on a single computer; the former is for
x64/AMD64/x86-64 EFIs and the latter is for IA32/x86/i386 EFIs. A 32-bit
EFI won't run a 64-bit binary and vice-versa. If you've got a disk with
both installed, they'd both be useable if you were to move the disk from
one computer to another, one computer having a 32-bit EFI and the other
having a 64-bit EFI. On the same computer, though, it just won't work --
at least, not with any EFI I've heard about.
The same thing follows through to the binaries that rEFInd launches.
That is, a 64-bit rEFInd, running on a 64-bit EFI, will launch a 64-bit
kernel, but not a 32-bit kernel; and a 32-bit rEFInd, running on a
32-bit EFI, will launch a 32-bit kernel, but not a 64-bit kernel. In
fact, rEFInd includes code to exclude the non-bootable kernel images
from its menu -- but this code works only for auto-scanned kernels, not
for kernels referenced via manual boot stanzas. Thus, my SUSPICION is
that you've created a manual boot stanza that references a 32-bit kernel
and that works on a computer with a 32-bit EFI, but that won't work on a
system with a 64-bit EFI. I don't know enough about your configuration
to be 100% positive of this, though.
Finally, GRUB is different. GRUB is a boot LOADER (with boot manager
functionality), whereas rEFInd is a boot MANAGER (which relies on the
kernel's own EFI stub loader to load kernels). GRUB can load kernels
across bit-depth boundaries, so a 64-bit GRUB can load both 64-bit and
32-bit kernels, as can a 32-bit GRUB. This is an advantage of GRUB over
rEFInd in some situations.
As a practical matter, the need to load kernels across bit-depth lines
is rare. It's mostly an issue on systems with 64-bit CPUs and 32-bit
EFIs, such some very early Macs and some modern tablet computers. Even
then, it may be better to stick with a 32-bit kernel. (This is what
Windows does on such computers.) Loading a 32-bit kernel on a computer
with a 64-bit CPU and 64-bit EFI is seldom beneficial; unless you have a
specific reason for doing this, I strongly recommend against even trying
to do it.