Hints: A Crossword

Getting started:
The clues are normal crossword clues, even if they look kind of weird. However, the grid is not a standard New York Times-style crossword grid (the clue numbers in those generally go up to way more than 27).
An Example Clue Solution:
Face of rock gem = FACET
Further starting hint:
This is a diagramless crossword in a cryptic-style grid. In particular, every square whose row and column are both even has a block in it. Additionally, Across clues always come before Down clues when they have the same number.
Stuck in the middle:
The clues are really kind of odd. It might be helpful to consider their lengths. This puzzle can be thought of as having two grids rather than just one.
Stuck on extraction:
What do the two grids have in common?