Hi, welcome to justanswer.com! My name is Mike.I am going over your question and I will be with you shortly.
That is software that allows your iPad to understand email addresses, phone numbers and websites, etc.. that's how you can click on them and do things. http://appleinsider.com/articles/16/08/08/inside-ios-10-apples-data-detectors-offer-quick-details-on-music-movies-actors
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What do you mean by MDM enterprise software? Just change all your iCloud, computer and device passcodes as well as email passwords then they cannot track you or see any of your communications...
There is no remote access software for iPads. Not possible. If you do as I suggested they cannot access anything. Good reading: http://www.macworld.com/article/1160313/iPad_security.html
You could also use a VPN service like this one: http://www.raynersw.com/netshade.php
By default, all iOS devices aren't Supervised. Devices can only be Supervised during activation of a new or fully erased device. With the Device Enrollment Program an organization can wirelessly enable Supervision on devices it owns. You can also manually Supervise a device by preparing it with Apple Configurator 2.
Here's how you can tell if a device is Supervised:
I can't connect remotely to your iPad only to a computer.
If your IT people did not setup your devices and you set them up yourself there is no way they could be managing them. If you don't see:
"This [iPhone, iPad, or iPod] is Supervised. [Organization name] can monitor your Internet traffic and locate this device."
You should not worry. In my mind if you don't trust your employees then they should be replaced.
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