Thanks for asking, I have been meaning to do more investigation as part of some programming tasks that I am working on and this is a timely inquiry. I have seen other questions in a similar vein and since the first answer missed the mark, here goes. (BTW, I tried an honest answer in the past and got voted down. Don't shoot the messenger.)
Sure, you can do the hide and show trick, but the real problem is that the Profile Info Web part (Microsoft.SharePoint.Portal.WebControls.ProfileInfoWebPart) is lame and limited. It will NEVER do what you really want. Admit it...you don't want to just show properties, you want to:
- Show the properties you want
- In the order you want
- With hyperlinks to the locations you want (like clickable email addresses)
- With a collapsible section containing the properties you want
In order to do that you have to do some programming. It's SUPER easy, In fact, you can do it all client-side, so it'll work in Office 365. I am working on a sample and if I can get past an issue with MDS I'll update this answer.