As I understand it ArcGIS Online has two types of accounts: user accounts (username/password) and application accounts (appId/appSecret).
When you invite a customer into a group you are always granting permissions to a user account. There is no way to invite an application into a group (at least not through the web interface). This means that an Organization (A) user can see content from Organization (B), but an Organization (A) application cannot.
BTW: I hope esri will consider changing this. But the reason for not doing so might have something to do with the business model around AGOL.
- Sune