The control files are however, different - they must be recreated when migrating from 32-bit -> 64 bit.
$ rman target /
Recovery Manager: Release 10.2.0.2.0 - Production on Sat Sep 9 12:23:50 2006
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connected to target database: ora10gr2 (not mounted)
Starting restore at 09-SEP-06 using target database control file instead of recovery catalog allocated channel: ORA_DISK_1 channel ORA_DISK_1: sid=159 devtype=DISK
channel ORA_DISK_1: restoring control file channel ORA_DISK_1: restore complete, elapsed time: 00:00:05 output filename=/app/oracle/oradata/ora10gr2/control01.ctl output filename=/app/oracle/oradata/ora10gr2/control02.ctl output filename=/app/oracle/oradata/ora10gr2/control03.ctl Finished restore at 09-SEP-06