Domino diamonds

You have a large collection of dominos, each labeled with two distinct symbols drawn from some infinite alphabet. The symbols have 180-degree rotational symmetry so dominoes can be rotated. In other words, these two dominoes are equivalent:

       +---------+---------+             +---------+---------+
       |         |         |             |         |         |
       |    N    |    S    |      ~      |    S    |    N    |
       |         |         |             |         |         |
       +---------+---------+             +---------+---------+

You wish to arrange your entire collection into 4-domino diamonds of the form

           +---------+---------+
           |         |         |
           |    Z    |    S    |
           |         |         |
           +---------+---------+
+---------+---------+ +---------+---------+
|         |         | |         |         |
|    X    |    Z    | |    S    |    N    |
|         |         | |         |         |
+---------+---------+ +---------+---------+
           +---------+---------+
           |         |         |
           |    Z    |    S    |
           |         |         |
           +---------+---------+,

where X, Z, S, N are distinct symbols. Actually, you have made lots of progress, having already assembled an infinite number of copies of each possible diamond. And more good news: the number of remaining dominoes is finite and a multiple of four!

By disassembling at most a finite number of your existing diamonds, will you always be able to arrange your entire collection as desired?