It's weird tho, it is not showing on my end regardless of which browser and phone combination I use. Even more so, it used to work even with CF enabled.
The Mobile theme still appears to be working on my end:
How does it work? I mean, does it redirect to a certain URL or container? Or does it read a header from the browser other than the user-agent data?
Jetpack looks for your user agent and automatically switches to the Mobile Theme. You can then go back to the desktop theme thanks to the switch at the bottom of the page. This switch will add a cookie to your browser,
akm_mobile, to force the desktop view in that browser. You can update that cookie by using the switch again, at the bottom of the page.
You can find out more about the Mobile detection here:
CF only have mobile redirection (subscription) to say m.domain.tld. A filter can be created but it is URL pattern based.
I'm not sure how the module works, but from what is available from CF, the redirection and URL pattern methods won't work, right?
There are no redirections to a mobile subdomain or to a specific mobile URL, so you won't be able to configure CloudFlare thoough these settings.
You're the first to report such problem with CloudFlare, however other Jetpack users have experienced similar issues when using other caching plugins, like W3 Total Cache or WP Super Cache. Luckily, these plugins offer options to handle mobile users and mobile themes like Jetpack or WPTouch.
I did notice, however, that you're using a very specific set up; when visiting your main site, a lot of the resources are served from another directory, jcsesecuneta.com/wb.
Do you use a CDN plugin, or manually host static resources in a different directory? If so, could you try to disable that plugin and let me know if it helps?