Assemble lozenges

You have a large number of 60° rhombi called "lozenges." Each lozenge has its edges marked with four distinct symbols drawn from an infinite alphabet. Lozenges may be rotated by 180° or reflected across either axis. A triple of lozenges forms a "hex" if the markings can be matched up in the following way:

     _____B_____
    /          
   /            
  C    E          A
 /                
/        ____D_____
        /          /
       /          /
  A    F          C
     /          /
    /_____B____/,

where A, B, C, D, E, F are distinct symbols. You wish to assemble all of your lozenges into hexes. Here are the conditions:

• You have access to a machine that can be programmed to produce arbitrary hexes; these hexes can be broken into new lozenges
• The total number of lozenges is a multiple of three
• For lozenge-type that you have, you have an even number of that type

Is it always possible to assemble all of your lozenges into hexes?