SAI-
I'm glad you're open to learning how different all GLOs are from each other! Even though it's not what you're used to, I think everyone agrees that no certain GLO is in the right while the other is in the wrong. Each just has different standards and beliefs on the issue.
Kappa, I believe, has the same regulations most other NPC groups have on how the badge is worn, specifically that it is worn on the left side. I remember, when I was on Panhellenic Executive, we always wore both badges to meetings, and the Panhellenic Badge was below and to the left. (To show you how silly I am, I even have it below and to the left of my Kappa badge on my badge pillow!

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Again, I think many good suggestions have been given: wearing only the immediate applicable badge to an event, wearing the solo badge and an honorary badge of the other organization or wearing one badge and some other sort of signifiaction of membership in the other organization. These are all wonderful suggestions and I think neither organization would feel slighted in any of those options. Hopefully if a similar situation does happen in your organization, you can give some helpful advice to the person!