I found an answer in Is Thunderbolt 2 to Thunderbolt 3 (type C) possible? that appears to solve my problem, but I am not "getting" it. (I tried to add a comment there, but I don't have enough credits to post there)
I have an late 2013 iMac with Thunderbolt 1 ports. I also have Apple's Thunderbolt 3 > 2 adapter (my wife's iMac is newer than mine, and it allows her to use older devices.) I have a J5create JCD383 Memory Card Reader/Writer with a Thunderbolt 3 male plug. I can plug that into my wife's computer, but not mine. What I'm trying to find is a way to plug that card into my computer, but I'm not finding any devices that will allow me to do it. The explanation on that other site says I have everything I need, but I don't see it.