Both teylyn and MLF pointed me in the right direction. I couldn't find the page "Manage my Features", to ensure Team Collaboration Lists was active (it was). Then it hit me.
When you create a site under Office 365, you get a web site:
and three site collections, including
UPDATE: After researching it further, I know understand, although it ticks me off and I can't justify it, I see what Microsoft has done. With Office 365, you get a very, very crippled website (the first URL) that you can allow anonymous access to. You then get all the site collections you want (starting with the second URL) that have full functionality, but cannot be made to allow anonymous access.
So what does this mean? It means you can't have a Calendar, Tasks, or even Links on the public web site. You know, because every organization or club that wants a website has ABSOLUTELY NO REASON to need a list of events.
So, the answer is, you have to make a custom list instead of a Calendar, and try to make it mimic the out-of-the-box Calendar that SharePoint has offered since at least WSS 2.0. Gosh, Microsoft, thanks for doing all my thinking for me. I have no argument with a Blank template. I have a serious problem with not allowing me to add a simple Calendar just because it's a public-facing site. I have posted on several Microsoft forums (Good Lord, they have a lot of different forums) but no one can give me a REASON for this, other than they don't want your public site to be a collaboration site. Okay, but a calendar is not just a collaboration tool.
Thanks to those who helped me here, but I need to post this as the answer for those who have the same issue.
UPDATE: I found an improvement on my original solution. Custom Lists don't have the view cabability of Calendars, even using an Event as the Content Type.
- Use SharePoint Designer to create a subsite (you can't do it from the browser).
- Create a Calendar in your subsite.
- On the home page of your subsite, remove all other web parts and add the Calendar.
- Add links back to your parent site, and link from your parent to the subsite home page.
It's not ideal, I can't inherit themes or navigation from the parent, otherwise it would be seamless to the user. But I'm not finished yet.
Thank you, yet again, Microsoft, for tying my hands. You are not making a better product, you are simply making more skilled Houdinis.