7 is considered a holy number. I assume you would agree. 3 has a way of showing completion in the bible. Like something 3 times. peter denied jesus 3 times, holy holy holy is the Lord, Father(1) Son(2) Holy Spirit(3), Jesus rose on the 3rd day,....etc
6 is just short of the holy 7. In six days god made the world and on the seventh he rested. Remember the Sabbath and keep it holy. By removing the sabbath day number the 6 becomes and unholy number. 6 Therefore can be said to represent toil, ambition, restlessness. It is expressed in its completeness by 666. 666 therefore is a very unholy number symbolically. Its putting emphasis on "coming up short" and ignores the fullness in 7. 666 is not so much counter expressive to 777 but a blasphemous assault on 7.
edit(forgot the point)
666 is a counter to 7. So 777 in turn is a counter to 666. Yes. Youve got it. It is a bit up a "one up" on the devil.