'Left' and 'Right' are not helpful terms here but the issue you raise can be phrased without them. The question is whether there is a logical connection between the politics that Heidegger espoused when the Nazis came to power and his philosophical work, principally in Being and Time (1927). What if anything joins political stance with philosophical opus? To deal with matters in as coherent order as I can:
Was Heidegger a Nazi?
There can be no doubt about this since the publication of Victor Farias' Heidegger and Nazism, Philadelphia: Temple University Press, 1989. And even the idea that Heidegger’s involvement with National Socialism was a brief, naïve dalliance which he came to regret does not stand up against the evidence. Heidegger remained a member of the NSDAP until 1945 and never recanted his connection with the Nazis. It is true that Heidegger became disillusioned with Hitler but he never repudiated the political vision which he thought he had found in National Socialism.
Alan Pascow aims to trace the origins and nature of that vision. He contextualises:
Nazi ideology as part of a broader cultural and historical outlook having important sources in early nineteenth century German Romantic poetry and philosophy. What needs to be better comprehended is German intellectuals' disenchantment with Enlightenment principles, which they feared to be issuing in a society defined by a rootless "mass man's" spirit of rationalism and technification, a will-to-power-and-domination penetrating every aspect of life.
I think that only by appreciating the lure of anti-Enlightenment thinking, which is so alien to most Americans and which therefore is most likely not properly understood, can we begin to make sense of Heidegger's conviction that Nazism had an "inner truth and greatness" that could inspire and transform all Germans. A new Germany, he believed, would in turn be able to lead all of the other nations of the world into a socially, politically, and spiritually superior postmodern era. (Ironically, Heidegger would have allowed for Germany the very will-to-domination that he condemned in modern society and with respect to which the new order was to be a corrective.)
Heidegger's enthrallment with this vision was so powerful that it sustained itself even after his disillusionment with Hitler, whom Heidegger after 1939 regarded as a "good-for-nothing" (Taugenichts) (assuming his wife's words to me in June 1988 are to be trusted). Indeed, Heidegger seems never to have lost faith in the possibility of a renewed and great nazified German nation, whose spirit would be articulated by his own philosophy as well as by the poetry of Holderlin and whose source would be uncovered in the language and culture of ancient Greece.
(Alan Pascow, ‘Heidegger and Nazism’, Philosophy East and West, Vol. 41, No. 4, The Sixth East-West Philosophers' Conference (Oct., 1991), pp. 522-527: 526-7.)
How does Heidegger’s politics connect logically with his philosophy?
This should be an open question. It should not be assumed that there is a logical connection but none the less a powerful case can be made that such a connection exists. I don’t think that Heidegger’s politics presupposes his philosophy or that his philosophy entails his politics. But there is a coherence between the two.
This is set out in an article by Emmanuel Faye, ‘Nazi Foundations in Heidegger's Work’, South Central Review, Vol. 23, No. 1, Fascism, Nazism: Cultural Legacies of Reaction (Spring, 2006), pp. 55-66. (GA = Collected Works, or Gesamtausgabe – ongoing.)
My research also deals with the 1920s, from Heidegger's lectures of 1925 entitled The current conflict for a historical vision of the world to Being and Time, published in 1927. 1 discovered the importance of intellectual bonds which linked Heidegger to racist authors and proto- Nazis like Erich Rothacker, Alfred Baeumler, Oskar Becker, and even the raciologist Ludwig Clauss, to whom Heidegger would confide: "what I think, I will say once I am a tenured professor." It is necessary to remain aware of this context in order to understand the affirmations of Being and Time such as the famous § 74 on historicity, in which Heidegger declares that existence is not defined as destiny, except through a community and a people. The identification of the authentic Dasein with Gemeinschaft and with the Volk is thus confirmed in 1927 in Being and Time. …
In addition, the lectures currently available from 1933-34 reveal to us that Heidegger, in his book on Kant from 1929, only re-addresses the question "What is man?" so as to transform it in his seminars and writings from the 1930s, into "Who are we?" He responds, "we are the people," the only people who still have a "history" and a "völkisch destiny." In effect, Heidegger understands this people as "völkisch," that is to say according to his own terms, as a race (Rasse). For him, it is necessary to accomplish a "total transformation" of the existence of man, in accordance with "the education for the National Socialist worldview," inculcated in the people through the Führer's speeches (GA 36/37, 225).
Can we seriously believe that for Heidegger these pro-Nazi views are only a fleeting political aberration that can be ignored in assessing the value of Being and Time? This would run counter to the most explicit affirmations of Heidegger himself. In effect in 1934, he explained to his students that "care - 'the most central term of Being and Time' - is the condition in which it is possible for man to be political in essence" (GA 36/37, 218). Heidegger declares at this time - one year after the National Socialist movement came to power - that "we ourselves," that is to say the German people, united under the Hitlerian Führung, are faced with an "even greater decision" than that which served as the origin of Greek philosophy! This decision, he specifies, "was articulated in my book, Being and Time." It concerns, he added, "a belief which must manifest itself through history" and concerns "the spiritual history of our people" (GA 36/37, 255). At the foundation of Heidegger's work, one thus finds not a philosophical idea, but rather a völkisch belief in the ontological superiority of a people and a race; moreover, the term völkisch designates in its Nazi usage the conception of a people as a marriage of blood and race, with "a strong anti-Semitic connotation," according to the Grimm dictionary. Frankly, an attentive reading of key paragraphs in Being and Time on death and historicity, with their celebration of sacrifice, of the choice of heroes and of the authentic destiny of Dasein in the community of the people, shows that this belief was already in place as of 1927.
With Heidegger, the question of man has thus become a völkisch question. It is in this sense that I spoke earlier of Heidegger's intention to introduce Nazism into philosophy. Of course, no true philosophy can align itself with the project of the extermination of human beings, a project to which the Nazi movement was committed. Therefore, I do not wish to say that Heidegger produced a National Socialist philosophy, but rather that he did not hesitate to utilize philosophical expressions such as "truth of Being" or "essence of man" to express something else entirely.
Left to myself I should say that Faye's final sentence undermines what he has previously said: if Heidegger used 'philosophical expressions' to express something else entirely, i.e. if he used them non-philosophically, then the continuity between his philosophy and his politics is broken. But I leave the sentence as it stands, for others to make up their own minds.