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What do you mean by a cross or a crossover?
Because your background in numismatics is an unknown to us, you may or may not be aware that the greater majority opinion in the community is that PCGS, NGC and ANACS are the top three TPGs (third party graders) in this order.
However, many feel, including myself, that the quality of the grading done by both PCGS and NGC are equivalent.
From my experience depending upon the particular situation PCGS slabs can sometimes bring around 10 - 15% more for the same coin compared to NGC in a retail environment.
Very seldom have I ever seen NGC slabs' pricing exceed those of PCGS and oftentimes they end up bringing just about the same price in the end.
On a side note I always used to mess up by saying NCG instead of NGC so I developed a dumb way on how to remember the distinction. Now when in doubt I just think, 'N'ot 'G'reat which are the first two correct letters of NGC even though the companies are identical in terms of quality control and professionalism etc.
To better clarify this issue here's a ten year old article to provide some historical perspective taken from
ebay archives addressing the overall subject matter of PCGS vs NGC that could influence your decision in the crossing over process:
http://www.ebay.com/gds/NGC-vs-PCGS...32811/g.htmlmdpmedia