No, they are different.
The .32 H&R Mag is, in short, a lengthened .32 S&W Long case loaded to higher pressures. Sort of like the relationship between the .44 Special and .44 Magnum or .38 Special and .357 Magnum.
You can shoot the shorter S&W Long cartridges in a .32 H&R Mag revolver, but if the chambers are in spec, the H&R Magnum cartridges won't chamber fully in a Long chamber.
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