Here's a 2015 letter (not from one of us) to the US magazine The Nation:
"I’m respectfully requesting that you retract the slur 'Trotskyite' in Richard Kreitner’s column on Bill Kristol. Kreitner incorrectly uses this derogatory word in reference to Kristol’s father, Irving.
"The difference between 'Trotskyite' and 'Trotskyist' is a bit like the difference between the 'N-word' and 'black' or 'African-American'. As I recall, the pejorative 'Trotskyite' was coined by Stalinists, casting Left Opposition followers of Trotsky as anti-Soviet fifth columnists and/or fascist agents. The slur 'Trotskyite' is thus a sort of etymological character assassination of the man and his supporters.
"The word 'Trotskyist', on the other hand, is a neutral term that merely refers to an adherent of Leon Trotsky. It does not deride or endorse those it refers to and/or Trotsky’s theory of permanent revolution, etc."
To which some words should be added.
The term "Marxism" was coined by opponents of Marx; the term "Luxemburgism", by denigrators of Luxemburg after her death; "Leninism", after Lenin's death by Stalin and others trying to claim Lenin's authority for ideas very different from Lenin's; and "Trotskyism", by opponents of Trotsky.
It was first used on a large scale in the denunciations of Trotsky and Luxemburg sent resounding through the Communist Parties as Zinoviev, Kamenev, and Stalin worked to tighten them up and make them "monolithic" after Lenin's death in 1924. The campaign was called "Bolshevisation".
Trotsky and his comrades of the Left Opposition rejected the idea that they had a special "-ism": they were just defending, and continuing as best they could, the basic ideals of Bolshevism on the basis of which the 1917 revolution had been made.
After a while, as with the term "Marxist", the comrades had to shrug and say: "Want to call us Trotskyists? Or Marxists? OK. We'd prefer a different term, but let's accept your term and move on to the substantive political arguments".
(The term "Stalinist", by the way, was put into wide usage not by our comrades or by hostile commentators, but by Stalin himself and his own followers, as they built up what would later be called "the cult of the personality").
"Trotskyite" came into wider usage a bit later, as the Stalinist terror unfolded. The first Moscow show trial in 1936 was labelled: "Case of the Trotskyite-Zinovievite Terrorist Centre".
In the denunciations of 1924, Zinoviev was one of the leading polemicists against "Trotskyism", and the arguments, however shoddy, were understood on all sides as one between different ideas within the revolutionary workers' movement. By 1936 Zinoviev was bracketed with Trotsky as the second demon of "the Trotskyite-Zinovievite Terrorist Centre". The Stalinists were denouncing "Trotskyites" not as mistaken comrades but as fascist agents. And in the following years they were murdering them en masse.
That is why "Trotskyite" has a different connotation from "Trotskyist".
The forerunner group of Workers' Liberty defined what Trotskyism means to us in the following words, in 1967:
"Trotskyism is the basic Marxist programme of the conquest of power by the international working class. It is the unfalsified Programme, method and experience of the Bolshevism of Lenin and Trotsky. It embodies the world experience of the workers' struggles' including the defence and development of Bolshevism by Trotsky and the Left Opposition in battle against the Stalinist counter-revolution in the Soviet Union.
"Trotskyism is the only developed working class alternative to venal Stalinism and supine Social Democracy. It means reliance on the self-controlling activity of the masses of the working class, which it strives to mobilise on the Programme of transitional demands as a bridge to the overthrow of capitalism and the attainment of workers' power. It is the Programme of the workers' revolution, organically linked with the practical struggle to aid its development.
"It is not only a programme, but the struggle to build a revolutionary party to fight for that programme.
"Its traditions are those of the Bolsheviks and the Left Opposition: workers' democracy, unremitting struggle for theoretical clarity, revolutionary activism, unbending hostility to and struggle against capitalism and those within the labour movement who stand for its continuation".
We'd still stand by that.
To us, "Trotskyism" is not a particular personal school of thought, but a broad political tradition, finding its origins in the very start of the workers' movement, in which Trotsky was an important leader and writer during a particular phase.
To us, the idea that "Trotskyism" could mean uncritical devotion to every particular view that Leon Trotsky had, or some belief that Trotsky said the last word on political issues, is utterly contrary to the spirit of real Trotskyism.
We've also concluded that since the 1940s "Trotskyism" in the general sense has divided into at least two distinct broad currents, one that called itself "Orthodox Trotskyism" and the other which could be called "Heterodox" or "Third Camp" Trotskyism. See here, for example. Our origins as a group were within a strand of "Orthodox Trotskyism", but we now believe that the "Heterodox Trotskyist" current offers better political anchor-points.