How do tugs push back the Antonov 225?
Quite simple ― use a (special) towbar! Aircraft with such unique front wheels (like the Antonov An-225 Mriya and Lockheed C-5 Galaxy) carry their own towbar around, which is then attached to the tugs for movement.

For example, here is the Galaxy's towbar:



From left to right: 68th Airlift Squadon loadmasters, Senior Airman Katy Mackey, Master Sgt. Kevin Foerster, Senior Master Sgt. Dustin Burleson, Airman 1st Class Brandon Stone, and Master Sgt. Alfonso Cervantes unload an aircraft tow bar off C-5A Galaxy upon arrival to Cairns, Australia on May 5, 2012. The tow bar was used airport officials to move the Air Force's largest transport aircraft in place to load the U.S. Coast Guard's HH-65C Dolphin helicopter. (U.S. Air Force Photo/Maj. Tim Wade)

And here's that of Mriya:



AN-225 towbar being removed from aircraft; screen grab from this youtube video.

And here's a Galaxy with a towbar:



Troops from the 439th Maintenance Squadron prepare to tow a C-5 Galaxy into Westover's Regional Isochronal dock, May 22, 2013, to test the newly-constructed mobile tail enclosure. Crews broke ground April 1, 2012, for the $5 million MTE, is a moveable structure on rails that will protect maintenance troops from harsh New England weather. The MTE is the largest structure of its kind in the Air Force. (U.S. Air Force photo/SrA Kelly Galloway)

This video shows Mriya's towbar in use.