My iCloud account doesn't have a calendar?

I use Google Calendar for my calendaring app. I am trying to solve a problem where calendar invites sent from other people using iCloud won't show up in my Google Calendar (I'm accessing gCal through Sunrise.am, but I don't think that's relevant).

I found this thread on support.apple.com which suggests if you have a Google account with the same email address as your Apple ID, then Apple will send the invite to your iCloud calendar instead of your Google Calendar, unless you log into iCloud and disable the "Use iCloud for incoming invitations" setting:

https://discussions.apple.com/thread/3454604?start=45&tstart=0

So, today I logged into icloud.com (which I have never used before) to try to disable that setting. However, I don't seem to have a calendar?

There is no mention of a calendar under the settings section. I also tried directly visiting https://www.icloud.com/#calendar and got redirected back to the home page. I've tried googling a ton, and can't find any mention of an iCloud account with no calendar.

I'm getting really frustrated with this, and just want invites from iCloud users to get through to my Google Calendar. I've already missed an important meeting due to this problem and really need it solved. :(

If you haven't signed in with your Apple ID on an Apple device at least once in its history, these are the only services available to you.

What is web-only access to iCloud?

Web-only access is a free level of iCloud service that's available to anyone, including people who don't own or use Apple devices. It includes access to Pages, Numbers, and Keynote for iCloud and 1 GB of free storage for any documents you create.

You can obtain the other features by signing in on an Apple device.

How do I get the full set of iCloud features?

If you have an iOS device or Mac, you can get the full set of iCloud features. Simply sign in to iCloud in Settings on your iOS device or System Preferences on your Mac. You'll get immediate access to all iCloud features and 5 GB of free storage. You can also choose additional storage plans from your device.

For more information, see this Apple KB article:

• Web-only access to iCloud