Eagle How to flip a symbol?
The new arrangement you're asking for on the opamp is non-chiral to the original. You can not get it by flipping and mirroring. The symmetry of the pins you want to swap is irrelevant in this case. If you want to swap the location of pins you need to edit the symbol. Right click the part in the schematic and click edit symbol. Then click library in the top bar and update all. 

You could also move the power and ground pins onto the axis of symmetry of the plus and minus inputs. Then you can flip all you want.

Note that either option will change this part for all schematics that use it. You'll have to make a copy of the part in the library.

EDIT:

For those parts which can be rotated and mirrored and rotated back to achieve the desired orientation you can write a script to make it a command. Here is a very crude, but working, example:

#usage "&lt;b&gt;Vertically Flip Selected Schematic Symbols&lt;/b&gt;&lt;p&gt;"
       "with parts selected - 'run ulp_file_name' &lt;br&gt;&lt;p&gt;"
       "&lt;author&gt;Samuel H.&lt;/author&gt;";

string ToDo="";
string New_Command;

if (sheet) 
{ sheet(S) 
 { S.parts(P) 
  { P.instances(I)
   { if (ingroup(I)) 
    { sprintf(New_Command,"ROTATE r90 %s ;",P.name);  
      ToDo+=New_Command;           
      sprintf(New_Command,"MIRROR %s ;",P.name);  
      ToDo+=New_Command;
      sprintf(New_Command,"ROTATE r-90 %s ;",P.name);  
      ToDo+=New_Command;     
}}}}}

exit(ToDo);


Unfortunately, if the Eagle ULPs can access the structures of selected items directly, I don't know how to do it. This ULP searches each part on the sheet and check if it's in the selected group. If it is, it adds the required operations to a string. That string is executed as a script upon exit. Just select whichever objects you want vertically flipped and run the ULP in the command line or as a hotkey item. 

Eagle is quite scriptable, admittedly it's sometimes like wrestling a thoroughly greased bear, but the job can usually be done. 

Incidentally you can write a script that will swap pins and save the part as alternate name part in the same, or different, library. It would be greatly simplified if it's the case you pointed out, where one is called '+' and the other is called '-'. That's a greaseless bear kind of problem.