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Boris Johnson in his book “The Churchill Factor: How One Man Made History” writes:

The only man to come through it with colours flying was Churchill’s parliamentary aide Bob Boothby, MP, who famously replied to Hitler’s megalomaniacal greeting of ‘Heil Hitler!’ with the only logical response: ‘Heil Boothby!’ said Boothby.

And he sources the information from Robert Rhodes James' book "Robert Boothby: A Portrait of Churchill’s Ally" (London: 1991), p. 138 (to which I have no access).

The phrasing suggests that Hitler was exclaiming "Heil Hitler!" himself when meeting with others.

Is this true? Are there sources or documents which confirm Hitler using this exclamation?

Short Answer

Despite the claims of some sites, I can find no evidence that Hitler himself said 'Heil Hitler', and it would make no sense for him to do so as the words were more than a greeting: they indicated obedience to the leader (Hitler) and were an oath of allegiance (to Hitler). This was made clear by a senior Nazi official, Gregor Strasser, as early as 1927.

The Boothby citation by Boris Johnson is wrong. Boothby's own account is that Hitler greeted him by saying his own name (without 'Heil') and lifting his right arm, and Boothby responded in the same manner.

When Boothby said 'Heil Boothby', it was in response to Hitler's secretary saying 'Heil Hitler'; he did not say this to Hitler himself, and Hitler did not say 'Heil Hitler' to Boothby.

Albert Speer, in his memoirs, relates that Hitler would sometimes respond to Heil, mein Fuhrer from favoured members of the inner circle with Heil, Speer (or Heil, Goering / Goebbels etc.).

Details

The citation by Boris Johnson is wrong (Wikipedia's Lord Boothby has it without the full context, making it potentially misleading rather than plain wrong). Boothby was actually responding to the 'Heil Hitler' greeting from Hitler's secretary, not from Hitler himself. Rob Baker in High Buildings, Low Morals on Boothby says:

in Germany he was once greeted by Hitler’s secretary with a ‘Heil
Hitler’, to which Boothby’s admirable response was ‘Heil Boothby’

Leslie Mitchell's biography Maurice Bowra: A Life clarifies further. Sir Maurice Bowra was a British academic and a friend of Boothby's from their time at Oxford University. Mitchell, citing both Bowra and Boothby himself, writes:

At some point, he [Bowra] and a party of journalists were given an audience by
Hitler where they were treated to an harangue on the theme that the
unity of Germany would not be political or economic, but spiritual.²⁴
It was on this occasion allegedly that Maurice replied to a salutation
of ‘Heil Hitler’ with a spirited ‘Heil Bowra’. Unfortunately, there is
no truth in the story, though it circulated widely and, as Maurice put
it, ‘brought me nothing but credit’.²⁵ What in fact happened was more
prosaic. Bob Boothby was greeted by Hitler’s secretary with a ‘Heil
Hitler’, to which he responded ‘Heil Boothby’. Later, as he
remembered, ‘the story was attributed to Maurice Bowra, who asked me
whether I would mind if he did not contradict it. I said I would be
delighted to share the honour with him, because I knew that if he had
been in my place, he would have done the same.’²⁶

Footnotes (24) and (25) are citing Bowra's own Memories (London, 1966). The source cited in footnote (26) above is Lord Boothby, Recollections of a Rebel (London, 1978). Here, Boothby gives his version of what happened when he met Hitler (not the secretary) - and it's missing the word 'Heil':

It is true that when I walked across the long room to a corner in
which he was sitting writing, in a brown shirt with a swastika on his
arm, he waited without looking up until I had reached his side, then
sprang to his feet, lifted his his right arm, and shouted ‘Hitler!’;
and that I responded by clicking my heels together together, raising
my right arm, and shouting back: ‘Boothby!’

From this context, it is apparent that Hitler was simply greeting Boothby and giving his name, unnecessary though it may seem. Of related interest, it was not uncommon in Jyland / Jutland (Denmark) even in the 1970s for adult men to introduce themselves to other adult males by simply saying their family name and nodding (no 'Hello' or such like, no first name).

Clearly, others (or even Boothby himself at times) have embellished the original version. For example, Israeli politician and journalist Yair Lapid's biography of his father Tommy Lapid has this:

Lord Robert Boothby…once told me the following story: “In 1931 I
visited the Fuhrer in Munich. When I entered his office he jumped from
his seat, stood at attention, raised his right arm in the Nazu salute
and shouted, ‘Heil Hitler’ I had no choice but to stand to attention
as well, raise my right arm and shout, ‘Heil Boothby’”

Who embellished this account is not in evidence; it could well have been Boothby himself for obvious comic effect.

Hitler's (various) physical responses when greeting people are well-documented (see Nazi Salute, photos and contemporary film footage). In addition to the salutes described in Wikipedia and seen in film footage, Hitler usually shook hands when greeting important or well-known people.

How he greeted them orally is less clear, but it most likely depended on the circumstances, the person or people and, quite possibly, his mood. As already stated, I can find no evidence that he used the words 'Heil Hitler' together, though this Quora post (unsourced) says otherwise (I've asked for a source so we'll see...). However, it would make no sense for Hitler to pledge loyalty to himself.

The ‘Heil Hitler’ greeting, Strasser wrote in January 1927, was not
only a symbol of personal dependence on the Leader, but contained in
itself the pledge of loyalty.

Source: Ian Kershaw, 'Hitler 1889-1936: Hubris' (1998)

Boothby's account is one way Hitler greeted people he hadn't previously met, but he certainly wouldn't have said his family name in this manner to children or to people he already knew.

Albert Speer's memoires Inside the Third Reich, relate that Heil Hitler was rarely used by the 40 to 50 people granted access to Hitler's 'afternoon dinner table' at the Chancellery; Guten Tag was more common. Hitler himself was fairly informal at these gatherings, shaking hands and perhaps enquiring about their well-being or sharing snippets of news. On the phone, Speer would end a conversation with Heil, mein Fuhrer', to which Hitler

sometimes replied, "Heil, Speer." This greeting was a sign of favor
which he only rarely vouchsafed to Goering, Goebbels, and a few other
intimates; underlying it was a note of faint irony at the mandatory,
"Heil, mein Fuhrer."

There is also no evidence in Hitler's written correspondence that he used Heil Hitler, but it was used by others sometimes (see, for example, this Martin Bormann letter).

Fans of old films may have been reminded of the line Heil Myself! in the 1942 satire To be or Not to Be, directed by Ernst Lubitsch, with Jack Benny playing a Polish actor who plays Hitler. Although wholly anti-Nazi, it was considered inappropriate by many at the time despite its Jewish director and star. The film is now considered a classic. Clips with Heil Myself! can be seen here (the first 20 secs) and here (1m 39 secs, at the end of the clip). The same line was also used in the Mel Brooks-directed 1963 film The Producers.

No, since the greeting is being made to him or referencing him as the leader

• he would not greet or reference himself

Nazi salute - Description

By autumn 1923, some members of the Nazi Party were using the rigid, outstretched right arm salute to greet Hitler, who responded by raising his own right hand crooked back at the elbow, palm opened upwards, in a gesture of acceptance.

...

The form "Heil, mein Führer!" was for direct address to Hitler.

The German version of the Wikipedia article also states:

Hitler hatte den Gruß aber auch aus seiner österreichischen Heimat mitgebracht, wo „Heil“ als Gruß unter Freunden verwendet wurde – und in den westlichen Landesteilen noch heute verwendet wird...

Hitler had also brought the greeting from his Austrian homeland, where "Heil" was used as a greeting among friends - and is still used today in the western parts of the country...

So if Hitler did ever reply, he would have used the single word Heil in return.

In the languages I am familiar with, someone, when being greeted, does not greet himself in return.

A source that claims otherwise, should be taken with a ton of salt.