You can say it, but only for yourself. The level of hatred from the right is generally beyond question, assuming you actually read/listen to these folks. And has been for a long time.
And that's the point where your "both sides" argument will struggle.
First, we're of a similar age, so you know that the tactic of labeling your political opponents as "sick," "traitorous," etc. is a very right-wing thing — thank you, Newt Gingrich! There was a level of politeness in our political speech until he stuck a knife into it, for which he was justifiably criticized on the right...until it worked. Then they made him Speaker.
Yes, Democrats have begun doing this as well, but I think they've shown remarkable restraint given how awful the GOP has been and for how long. Angry, nasty liberals have always been out there. But unlike conservatives in the GOP, they've not been in the upper echelons of the Democratic Party.
As for retreating, conservatives have long been shut into their own world, be it Rush Limbaugh, Fox News, whatever. Arguing with a conservative has been like taking hallucinogens for decades. You haven't lived until a relative says "you must hate America!" for suggesting our nation's foreign policy might affect how we're viewed worldwide.
As for engaging with the other, don't you recall the 90s and 00s when the standard Republican line was "never compromise"? It wasn't subtle, they said it out loud. Often. Meanwhile, Democrats are still out here compromising, and it's been damn near 30 years. The left is Charlie Brown, and the right is Lucy with the football. Over and over and over and over...
The other is a question of how the labels apply.
There is no evidence that the left generally or the Democrats specifically are "Communists!" or "Atheists!" or any of the other colorful epithets applied by the right. The anger has always been disproportionate and inaccurate.
But how inaccurate are the charges made by the left, actually?
Homophobia is inarguable. The American right is and has always been loudly, proudly, and often violently homophobic.
The charge of misogyny is also impossible to escape from. My parents were active in trying to pass the ERA back in the 70s, so I've been hearing it for a long time. The only difference is people feel freer to call it out.
I don't know how the science denial charge goes wrong, either. I mean, science has various things to say, and the Christian right denies a lot of it. What else would you call it?
The comeback on these points is usually something about religious belief, but that doesn't fly. There's a disproportionate response problem.
For instance, the left doesn't want to force anyone to be gay or require churches to hold gay weddings. The religious right has always had its boot on the necks of the LGBTQ+ community. The left doesn't want to reverse that, they just want you guys to stop.
The left doesn't want to repeal the Second Amendment, or ban the reading of Bibles, or any of that nonsense. They think science books should be full of science, and the First Amendment should apply to all religions, not just Christianity.
The upshot seems to be that the left describes oppression as one group trying to force its beliefs on others, while the Christian right describes oppression as preventing them from forcing their beliefs on others.
It's not balanced. Both sides do it, but at some point the quantitative becomes qualitative.