Besides the hard reasons of oil routes which brought neocons and neoliberals together in the Balkans and Chechnya and into Asia was the way they view the future.
Both groups want one world hegemony and the end of the concept of sovreignty and borders (remember the statements of Clinton's Strobe Talbott?).
The neo-liberals want this to happen under the flag of the UN with the USA and NATO playing important roles as the UN's police force. The Neo-cons also want ths agenda but under the American flag with NATO serving as America's foreign legion for this one world order or American hegemony.
That is why Cook went.
One important difference is that the Leftist, what you call neo-liberal, new world order would be fundamentally anti-competitive and anti-freedom whereas the neocons seek a world safe for US-style capitalism. (Compare, for example, the French vision of the EU with NAFTA.) Also, I think the NATO role will be very limited going forward as a result of the fractures displayed in this crisis, including as regards the treaty obligation to defend Turkey.