Why do my invites end up in Google Calendar instead of iCloud?

Cirrux sends invitations the standard way: an email with the event attached as a calendar file (iMIP, text/calendar with method=REQUEST). Every calendar app that accepts invitations by email understands it.

Why iCloud invitations behave differently

When someone invites you from an iCloud calendar, iCloud first checks whether your address belongs to an Apple Account. That includes a Gmail address you use as your Apple ID or added to it as an extra address. If it does, iCloud puts the invitation straight into your iCloud calendar on its own servers and sends no email at all.

Nobody outside iCloud can do that. Cirrux, Google Calendar, Outlook and every other calendar can only email the invitation, so it goes to whoever runs the mailbox. For a Gmail address that is Gmail, and Gmail adds it to Google Calendar. Apple offers no way for other services to deliver into an iCloud calendar, so this is not something Cirrux can change.

What you can do

• Give out your iCloud address for invitations. Invitations to your @icloud.com or @me.com address arrive in iCloud Mail, and Apple Calendar picks them up from there.
• Open the invitation in Apple Calendar yourself. Turn off "Add invitations to my calendar" in Google Calendar's settings, then open the attached invite.ics from the email to add the event to your iCloud calendar.
• Show both calendars side by side. Add your Google account to Apple Calendar on your iPhone or Mac, so events that land in Google Calendar show up next to your iCloud ones.