I am somewhat to the right politically of many of my online friends - this isn't entirely surprising as many of them are academics, where I have a business background. But that doesn't make me a conservative with a small C. In fact those fairly close to the centre of politics on either wing are probably least likely to suffer less from conservatism on the matter of equality versus culture and religion than those who sit firmly on one side or another. Right wing conservatives want to preserve their own culture, while left wing conservatives want to preserve everyone else's culture but their own, probably due to an existential guilt over the imperialist past.
However, I truly can't understand how we justify the way that we unthinkingly put religious and cultural demands above equality. Who decides which should have the upper hand? You can see why, in the past, when a particular religion had a huge hold on a country this might the case, but should that still apply in the 21st century?
So, for instance, do Catholics and Muslims really deserve the right not to allow female priests or imams? Is it acceptable that Church of England vicars won't perform same sex marriages? Note that to question this is not in any sense a matter of suppressing religion. I'm not saying that people shouldn't be religious or should be prevented from practicing a religious faith - just that it's not clear why following a religion gives you the right to overrule aspects of equality that are broadly accepted by society. We already limit many of the practices of religion (stoning adulterers, for example), so it's not clear why this particular aspect gets ignored - unless it's that the establishment is conservative with a small C.
This move to equality extends beyond religion to wider cultural applications too. Why should Masons, for example, be allowed to prevent women from joining their lodges? (Not that I can imagine many would want to join.) The only reason I can see that we allow culture and religion to have an override on equality is unconscious conservatism. And perhaps one good thing that could emerge from an era of political upheavals is that we can re-examine assumptions like this.