I would like to link from my LinkedIn profile to my Stack Overflow profile, and vice-versa. Is this possible, and if so how?
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No, this is not possible.
Stack Exchange uses Open ID for authentication. LinkedIn uses their own system.
The two have no way to communicate.
Yes, this is possible. Communication per se is not necessary, just reciprocal hyperlinks in your SO and LinkedIn profiles.
As far as specific profile fields:
You can see below that I have done this with my SO and LinkedIn profiles.
Figure 1: Link in SO Profile to LinkedIn Profile
Figure 2: Link in LinkedIn Profile to SO Profile