Thanks for the good question. I've been looking around for evidence for this point view. I think there is truth to the argument but I am suspicious of the source.

Let's discuss this source a little bit. It was posted on a website called "Militant Islam Watch." From the conclusion of the article:

Today, in the twenty-first century, Fascists, Islamist Fascists and Communists are again forming alliances against the forces of freedom and democracy. But the Islamo-Fascists may be stronger than the Marxists, the Socialists and the Communists.

It is plain the author has an agenda. Now, that doesn't make everything he has said totally wrong, but it does invite us to put some pressure on his argument. While he acknowledges that the Nazis cracked down on the KPD in the end (more on this), the following interpretation is too broad of an interpretation of the facts:

There is a psychological mechanism that somehow draws adherents or followers from completely contradictory ideologies and movements together in a common struggle against freedom and democracy, indeed against the West.

Yes, the KPD and the Nazis shared a fundamentally anti-liberal, anti-democratic outlook. They were aligned in despising the Weimar Republic. As Otto Friedrich put it, the KPD and Nazis together represented "the bitter nay-saying of a majority of the German population." Anti-democratic, yes, and well-observed. Anti-western? Pointing to an alliance of "Islamic-Fascists and Communists?" I have no idea what he is talking about there, and he has made an interesting discussion of different anti-democratic outlooks needlessly ideological.

Provided that the source above is good - I don't have the book (Franz Borkenau's World Communism) to hand, and frankly have never heard of it - the KPD despised the SPD, bitterly, inexorably, and at times beyond all rationality, and it does seem that this attitude had to do with Comintern policy. Shirer corroborates, unfortunately without a citation:

The Communists, at the behest of Moscow, were committed to the last to the silly idea of first destroying the Social Democrats, the Socialist Trade Unions and what middle-class democratic forces there were, on the dubious theory that although this would lead to a Nazi regime it would be only temporary and would bring inevitably the collapse of capitalism.

And, yes, as early as 1914, when the SPD voted for war, the Communists had begun seeing them as "social-traitors," and they became the "social fascists" when they opposed and helped suppress the Communist-Socialist government of Saxony in 1923. But, interestingly, both parties accused the other of supporting Hitler during the 1932 presidential election, the SPD accusing Thalmann of supporting the Nazis, the KPD accusing the SPD of supporting Hitler through the conduit of Hindenburg. There was, on the other hand, a SPD-KPD Anti-Fascist league formed in May 1932, already too late to make much of a difference.

So, it's important to emphasize the KPD had very specific reasons, not all of them unjustified - the SPD-led suppression of the various Communist risings in 1918-1919 and then again in 1923 were bloody indeed - for fighting against the SPD in particular. Many of their actions, seemingly taken against democracy or in support of Nazi policy, must be seen in the light of this rivalry and enmity.

Though other major facts check-out, it is at times a threadbare argument. These are the facts the author cites:

-The 1931 Prussian plebiscite, during which the KPD joined the Nazis in opposing the SPD government of Prussia.

-The KPD and Nazis voted together in 70% of cases in the period of 1929-1933. But this is a potentially misleading statistic, as there are times when the two parties were joined by others. The Nazis sometimes voted with other parties in the Reichstag, though the votes were typically negative ones. They joined the SPD in rejecting Bruning's emergency decrees for reforming German finances, for the SPD a rejection of Bruning's anti-democratic methods, and for the Nazis a play at encouraging instability. But the Nazis could always be counted on to vote against a sitting government.

-The KPD joined a Nazi motion of no confidence in October 1930.

-Both Nazis and KPD heckled Socialist Herman Muller in 1930.

-The 1932 transit workers strike, when Communists and Nazis shut down Berlin's transportation system.

-A long section entitled "Paving the Way for Nazi Rule," which does not provide evidence for the argument implied in the title. I suspect the author shies away from discussing the parliamentary maneuvers of 1932 since there were at least two broadly based coalitions including Nazis at this time, both against Presidential-decree governments. So there was certainly not an exclusive KPD-Nazi cooperation in the late Republic's defining votes. Most of the political spectrum in the Reichstag joined the anti-Bruning coalition of April 1932; many more joined the anti-Papen coalition of September, 1932, which was a vote of no-confidence supported by the Centre, SPD, KPD and NSDAP. By the end, the bureaucrat Bruning would be regretted, even compared with the Papen government to follow. In retrospect we might see these historic votes as signposts on the road to Nazi rule, but the SPD at least was motivated by a legitimate hatred of the policies and methods of Bruning and Papen.

What I don't understand about this article is why the author seems to avoid the evidence that seems most likely to make his case. He is trying to argue that the main weapon of the KPD was political skullduggery - joining the Nazis in strikes and in certain votes, for instance. There is a whiff of conspiracy theory about all this. He is trying to show that there is a communist-fascist-Islamic[???] political conspiracy afoot even today. If he had only been true to his argument about the "totalitarian mindset," he would have raised the widely acknowledged fact that a huge number of SA thugs and even later party officials learned politics and street violence in the KPD. The SA and other party organizations eagerly took up former communist party members. Unless you had been some high-ranking official, in which case it would be hard to scrub your history clean, the Nazis were happy to take on another thug or grunt. And this absolutely has to do with a shared psychology of violence and anti-liberalism, a hatred of compromise and moderate government. I think this sharing of manpower is the most important support of the argument this author tries to make, but he avoids it because it complicates the idea of a political conspiracy: the real overlap was on a sociological level, the willingness of angry young men to switch allegiances between the parties.

Finally, the author does make passing mention of the fact, but it's worth emphasizing that Nazism was touted as an anti-Bolshevist movement and accepted as such. Hitler's promises to deal with the Commuinsts and Socialists, especially in the wake of the Reichstag fire, demonstrate the superficiality of their earlier political collaboration, such as it was. So, that particular quote you gave may indeed be too strongly worded. As the KPD rhetoric during the 1932 elections proves, Communists did see Hitler and fascism as the enemy. They hated the SPD concurrently. I don't see why it's necessary to argue they hated one more than the other.

In sum, I'm not convinced that the Nazis would not have been able to gain political power without KPD support, though it is beyond doubt that KPD policy did aid the Nazi movement at various points.