Answer: 630

Solution:

Choose the coin orientations and the gold/silver positions independently. Record the orientations from bottom to top as a string of U (engraved face up) and D (engraved face down). Two adjacent coins are face to face exactly when the lower coin's engraved side faces up and the upper coin's engraved side faces down — that is, exactly when a U is immediately followed by a D.

A string of U's and D's avoids the pattern UD exactly when every D precedes every U, so the string is $\text{D}^i\text{U}^{8-i}$ for some $i = 0, 1, \ldots, 8:$ there are $9$ allowable orientation strings. Independently, the gold coins occupy $4$ of the $8$ positions in $\binom{8}{4} = 70$ ways.

The total is $9 \cdot 70 = 630.$