?- I would do this.
I assume that your system is in a tower and that you can come to the cables connecting the HDD controllers to your HDD.
I would not want to do anything that wrote data to that swap partition, including running any form of linux while that swap was present, also including rescue and restore CDs.
However, were you to disconnect the HDD containing H, and use a copy of PartedMagic or some other linux live-cd with the ability to refresh and re-find devices such as has GParted in PartedMagic, linux would boot without finding and using the swap partition.
After linux has booted, connect the other HDD, refresh the devices, open a terminal, and use the command <fdisk-l> (list) to read the partition tables and post the information here. See man fdisk. You might use the same avoidance method with any other program on a live-cd.
If it is a problem burning an ISO image in Windows, you can burn PartedMagic with several burner programs. I have used this.
Before trying any recovery operations, I would use Clonezilla or the command <dd> to make a copy of H to a partition or device that I could use as a work copy. I would not want to try working on the original, nor would I change the existing swap/H in any way.