I’d argue quite strongly that liberalism is a moderate right to center position nowadays, so that sounds about right. So saying that the liberal parties are classified as right in such an analysis doesn’t seem like a contradiction or counter argument to me.

Frankly, I have no idea why people consider liberal to be left wing. How else would you describe an ideology that emphasizes private property, free trade, and market capitalism as anything other than at least moderate right? Sure, if you’re a monarchist that whole “equality before the law” sounds radical left, but for the rest of us… And that’s before we talk about neoliberalism and it’s emphasis on free markets and minimal government interference, which is hardly a leftist position.