Left vs right is working people vs billionaires. It is right-wing propaganda trying to deny or obfuscate that. That is why socialism is left-wing and free-market neoliberalism is right wing.
It is not just saving money on taxes, but having desperate old people as cheap labor and pre-retirement people as customers for annuities. Particularly as desperate, cheap labor, since that helps push wages down for any sector that they could work in.
rainbow capitalism is mostly marketing towards polite liberalism. Homophobia/transphobia is their attempt to pit people against each other. Just like racism, it is not that these things didn’t exist before capitalism, but the exacerbation and use of it to divide the working class is what capitalists and reactionaries use.
No, they use the culture war to divide and confuse people. Not because they want minorities to be respected, but to direct hate to gay and trans people, rather than them.
Fascism and neoliberalism is right-wing and market based. Again, terms are purposefully obfuscated and disingenuously misused, but fascism is about privatization and deregulation (Privatisierung, in the original Nazi). While technically some working class are part of reactionary movements like MAGA, it is disproportionately either the Lumpenproletariat or not working class but what Republicans pretend is the working class (reactionary small business owners, who are working class because of some cultural marker like wearing blue jeans and liking country music or hating woke). We live in a liberal, capitalist system. So socialism is almost always, by definition, left wing and pro-capitalist and pro-billionaire politics is almost always, by definition, right-wing.