@misty TL;DR: There's a very high probability that any two serial ports will plug together with "standard" cables and adapters, the only _likely_ bugaboo is if one end requires hardware handshaking and either the cable or the other end does not support it. You can sometimes "fix" this with some creative jumpering.
@elb Thank you so much! This is really helpful. I found a copy of the pinout for my device, which specifies that it does use the handshaking signals and not just the data pins/ground, so I’ll need to make sure whatever I buy supports that.
@misty in particular if you want to use a USB serial adapter you'll need to verify that it has hardware handshaking. The physical 9 to 25 pin adapter is _probably_ fine.
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