Defining political left and right
Although it's normally advisable to steer clear of words that mean different things to different people, and especially of words that don't map to a single concept, political "left/right" is too ubiquitous to avoid using it in discourse, so I try to use it in the most meaningful way that still approximates how others use it.
There are two prime ideas that constitute leftism to me:
Extreme prioritarianism. Although not necessarily consciously consequentialist, leftists constantly put aside freedom of association to focus on inequality; hence why they tend toward communism.
There are no moral values outside of happiness and peace. Leftism rejects all other standards of judgement, including the concept of degeneracy as it exists in eg. conservatism, and often also rejects objective standards in amoral domains like art and game design. It's also why "just your opinion" culture is associated with leftism.
It's interesting that the second idea is like a version of the libertarian NAP applied to voluntary interaction instead of just the use of force: not only is it wrong to initiate force, it's wrong to initiate peaceful negativity of any kind. And correspondingly leftists often conflate violence with non-violence, especially in the context of bigotry.
I consider anyone with these two traits more or less leftist. Left-anarchists usually insist that believing in "capitalist" property rights makes you right-wing regardless of anything else (so they see eg. the democratic party as right-wing), but I disagree because I see the rejection of property rights by the far left as merely a consequence of #1 rather than an inherent aspect of leftism, and members of this "establishment left" faction do show a high degree of prioritarianism.
Rightism is much less well defined; lots of people who don't identify as right basically use it to mean everything bad (far more than the number of right-wingers who do this with leftism). Most people who try to define rightism will base it on a selection of two polymorphic values: "tradition", and "hierarchy". Obviously tradition is useless as a definition for ideology because it depends entirely on what traditions. Hierarchy also is never a value, it only seems like one through leftist eyes because leftists focus so much on their vision of equality that they read its opposite into their enemies as a value.
In practice the traditions referenced are Judeo-Christian ones, which from a philosopher's point of view is a totally arbitrary grouping of traditions, but since they cluster together so much and are a point of identity to so many people, it is sadly useful to have a word for this other than Judeo-Christian (since many atheists subscribe to it).
But I prefer to call that "conservative" rather than "right-wing". I use "progressive" to mean the opposite of these traditions. Conservative and progressive relate only to aspect #2 of leftism above, not to #1. I guess the logical conclusion is to use rightism to mean a combination of conservatism and an opposition to that #1.
Note that both tenets of leftism are actual principles, but conservatism is not. Leftism is based on fundamental aspects of human nature, which is universal and immutable. Conservatism only exists because Judeo-Christian religion and history exist; there is no fundamental fact about human nature that gives rise to conservatism.
An interesting reflection is that as it is not anchored to anything timeless, conservatism cannnot be timeless. There will be a time after conservatism, but there will never be a time after leftism.