To be fair, most Twentieth century political categorisations are now quite challenged. Fascists masquerade as Conservatives and Socialists and Liberals have somewhat been mixed up. Even the Left and the Right could not keep their Estates Generale purity: The Left sat on the right so long that they forgot their place, and the Right stole so many Left ideas that they dropped the label, at least when they referred to themselves. And, even in this melee, Neoliberalism stood out - perhaps just because it is so ubiquitous, so temptingly vaguely and yet so decidedly influential.
I could say that I can't define Neo-liberalism but know it when I see it, but I read The Economist every week, and therefore, see it ever so often. Therefore, despite the difficulty, I shall try to 'symptomise' what Neoliberalism stands for, or, how to know a neo-liberal when you meet one. And, I would try to do so using a model based on four parameters - Globalisation, Inequality, Regulation and Society - which I personally use as a shorthand to understand political categories.
On Globalisation, Neo-Liberals are globalisers and believe in free movement of capital and free trade. This is one of the founding tenets of Liberalism and the neo-liberals haven't strayed further from their base. And, this is why Europe, rather than Brexit, is a neo-liberal project. The neo-liberals are also usually committed to free movement of people, but, from their corner, it is to be justified by an economic rationale rather than anything else. Compassion for refugees is not a neo-liberal thing, but the business' needs for skilled workers very much is.
On Inequality: Neo-Liberals are free marketeers and therefore, they believe Inequality is an economic phenomena which appears and disappears with changes in technology and the production process. So, for them, this is a natural event and one can't legislate it away. Rather, the best way to deal with inequality is to encourage more efficient markets to do their job. This does not, however, mean that the Neo-Liberals do not favour intervention: Their interventions are more likely to be oblique, through improvements in Education and Health, rather than taxation and welfare.
On Regulation: The Neo-Liberals have at heart 'Government is the problem' mantra, so that makes them different from old-style Liberals, who loved intervening in one way or another. But the lack of appetite for regulation doesn't make them free-for-all Libertarians, as they believe in expert knowledge and the right of the few enlightened individuals define what happens to rest of us. This primacy of the professionals is also at odds with the Conservatives, who have a natural affinity for tradition (which professionals often seek to rationalise and overturn) and tacit knowledge (which professional knowledge discounts and replaces). So, 'nudge, nudge, nudge' is the neo-liberal mantra, which, in more traditional language, would be called 'Carrots and Sticks'.
On Society: If Reagan provided the Neo-Liberals the catchphrase for Regulation, Thatcher gave them the definition of society:"There is no such thing as Society". What we call 'society' is a collection of self-interested people cooperating with each other for pure reasons of survival and advantage. Traditions are merely an illusion, and membership of this 'collective' is based on economic roles and contributions, which makes it somewhat easy to make the case of immigration. The neo-liberals, therefore, wouldn't approve of beef-crusade of Mr Modi in India but would solidly stand behind him when he scraps environmental employment regulations.
In summary, then, the Neo-Liberals are a hopeful bunch (whereas the Conservatives are pessimistic) who believes in an economic community (again, ad odds with Conservatives but more like the Marxists) where free markets and individual initiative can drive progress and maximise common good (this is not what Socialists or Marxists would accept). They believe in unregulated development of technologies (Conservatives wouldn't) and free global movement of people and capital (Conservatives and Socialists wouldn't like that, despite the latter's insistence on solidarity of men). Neo-liberals are less hot on regulations than their Liberal predecessors - this is what justifies the 'neo' tag - but prefers intervention in terms of health and education. They are a peculiar late Twentieth Century phenomena who we have come to denigrate, but only because almost all of us, unless you are landed gentry or dyed-in-the-wool communist, have all become neo-liberals ourselves and feel the need to denounce the ideology which stems from everyday compromises that we do (and believe we are doing it for pragmatic reasons).