That is a modern gun made for .32 H&R Magnum. There is no doubt, none whatsoever that if it is safe to fire .32 H&R Mag, then it can safely fire .32 S&W and .32 S&W Long.
However, let it be known that in my own personal opinion... there is little reason to do so unless someone dumps a pile of that on you or it's all you can find. Both the .32 S&W and the .32 S&W Long are so light & weak that they almost feel like a misfire when you light them off. No offense intended to any huge fans of those "both feet in the grave" cartridges.
Not that the .32 H&R Mag is a barn-burner... it is not. It can be handloaded to be a barn burner if you so choose.
If I were not a handloader, I'd be looking to swap out that revolver because ammo is likely to be a hassle and pricey. However, since I am an avid handloader -- I'd be measuring the length of that cylinder and punching up Hamilton Bowen on the speed dial and see if he'd be in the mood to ream those chambers to .327 Federal Magnum.
One other option that's even better if you aren't a handloader: Use
this revolver as the final straw that makes you
become a handloader. It's a fine cartridge to learn on, you'll add a wonderful new dimension to your firearms joy and you'll be able to feed that great wheelgun without breaking the bank.
If you think you'd like to investigate handloading and how to get started, venture over to the Handloading & Reloading area of the site.