To answer the title question, it is an unstable compound of both, which is a large part of what makes it both so powerful (i.e., attractive as an ideology which can appeal strongly both to reason and emotion) and so dangerous (liable to lead to disaster and factionalism, which, when communists actually get their hands on state power, leads to vicious purges).
Fascism is also such a mixture, though a lot heavier on the Romanticism, and the Rationalism is largely confined to instrumental uses, such as organizing the bureaucracy of the death camps.
The difference between communism and fascism, however, does not have much to do with this. The difference lies in their diametrically opposite view of the relative moral worth of human beings. The left, from revolutionary communists to democratic socialists and American so-called “liberals,” believes, at root, that all people are of equal fundamental worth, and equally deserving of respect, and of the good things in life. Most leftists can accept that people are unequal in terms of their abilities and character, but they do not think that that is relevant to what people deserve out of life. Just because you were lucky enough to be born, or made by life circumstances, into someone who is smart and industrious, it does not follow that you deserve more happiness than someone who was born, or made by circumstances, to be stupid and lazy.
Also, from a leftist perspective, if people do evil things, it is not because they are inherently evil (or any more so than anyone else), but because evil circumstances forced them into it or warped their minds. To minimize the amount of evil that happens, society needs to be re-structured so that such circumstances arise as rarely as possible. Luckily (as most leftists believe) the sort of society in which this would be the case is precisely the sort of egalitarian society that leftists think is the most just anyway. If everyone has enough to meet their needs, and nobody has ostentatiously any more over and above what they need than anyone else does, then nobody will have ane reasons of either necessity or envy to try to take away the things that are needed by others.
By contrast, the right, from Nazis to democratic conservatives and the American “center,” believe that some people are fundamentally more worthy than others, and deserving of more respect and more of the good things (and, usually, that some people, at the other end of the scale, are virtually worthless, and undeserving of anything good). The supposed basis of this inequality may be different in different versions of right-wing thinking. It might be race or nationality, it might be “family,” it might be intelligence, or industriousness (see Magiver’s post above for an example of this sort of thinking), or courage (such as when we are told that wealthy businessmen or stock traders deserve their riches because they “take risks” that ordinary folk like miners and construction workers do not), or it might be just a matter of some people being born leaders, and some being born followers (and, of course, it is also often various combinations of these things, and probably others, in different proportions). However, whatever the basis for it, the fundamental idea is that some people inherently deserve admiration and reward more than others.
From the right-wing perspective, evil happens either because some people are inherently evil (they cannot transcend their evil nature, so the only way to contain their evil is to imprison them or, if you are truly rational about it, kill them - this explains both Nazi death camps, and the Texas justice system), or else it happens because some people have got into the wrong place for them in the necessary social hierarchy. If a natural leader is (like, say Napoleon) is born into humble circumstances, or if a natural follower is born into power and privilege (like many incompetent hereditary monarchs seem to have been), bad things tend to happen unless and until they are reassigned to their proper level. In order to make such evil circumstances less likely, some more “liberal” right wingers (especially if they do not believe that race or family history are particularly important factors in determining moral worth), argue for maximizing equality of opportunity, so that the more talented, industrious, courageous or whatever will generally tend to rise to the level of power and privilege within the society that is appropriate for them, whereas the untalented, cowardly and slothful will sink to their own proper level.
Despite some superficial similarities, this sort of right-wing egalitarianism (which only some segments of the right adhere to anyway) is very different in its aims and effects than is the left-wing egalitarianism (which is universal, and fundamental, on the left). Left-wing egalitarianism seeks equality of outcomes, with everybody, ideally, ending up with an equally good life in as far as that is humanly achievable (of course, leftists disagree enormously amongst themselves about the best way to reach this desirable situation, and, indeed, about how far it can truly be achieved). For right-wing egalitarianism, by contrast, equality of opportunity is a means to the end of creating a radically unequal society, that will be maximally stable and conflict free because it maximizes the chances that everyone will find their proper level within it.