Is there any term for someone who pretends to be neutral (in discussion), but in fact supports one of sides?
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A wolf in sheep's clothing “is used of those playing a role contrary to their real character” (per wikipedia) and may be appropriate or may overstate the case, depending on context.
An impostor is “Someone who attempts to deceive by using an assumed name or identity”; the relevant identity in this case being that of a neutral participant, a participant with no vested interest, etc.
An adjective for behavior mentioned in the question is disingenuous, meaning “Not ingenuous; not frank or open; uncandid; unworthily or meanly artful”. (Ingenuous itself has among its senses “Straightforward, candid, open, and frank.”) Duplicitous, with synonyms like deceitful, double-dealing, and two-faced, also is relevant. (Synonyms of two-faced include deceitful, hypocritical, duplicitous.) Adjective specious, “falsely appearing to be fair, just, or right: appearing to be true but actually false” also applies.
Also see: A word for a worldly wise person who pretends to be naïve?, What's single word for fake image? [with links to words like two-faced, Janus-like, deceiver, fake, double-dealer, duplicitous, polymorph, chameleon, wolf in sheep's clothing, poser, duplicitous, back-stabber, hypocrite, fraud, imposter, poseur, whited sepulcher, Devil in disguise et al], What is a word to describe a person who wears several masks?, What is an idiom/slang for "someone who pretends to be good when they're not"?, Common phrases for something that appears good but is actually bad, etc.
In some situations, the terms fifth columnist (“someone who belongs to a group that secretly undermines another organization”) and sleeper (“A spy, saboteur, or terrorist who lives unobtrusively in a community until activated by a prearranged signal; may be part of a sleeper cell”) may apply.
I'm going to say mealy-mouthed which is related to speech. It is often used to describe media who tries to convey an idea with an evasive and indirect speech to pretend to be neutral but there is often a hidden agenda or media bias.
In technical contexts and psychology, there is a term pseudo-objectivity which refers to the false sense of objectivity but also defines something that is falsely conceived to be objective. So there is this confusion between objectivity and subjectivity of the source.
I think we would usually call this kind of person very diplomatic.
ODE defines this in the following way:
An answer from ask (dot com) as to what a diplomatic person is:
Diplomatic people often keep their opinions to themselves to avoid confrontation but are actually quite opinionated.
Biased would fit well, except it doesn't convey automatically the idea that the participant is hiding his intent.
Predisposed and its synonyms convey the meaning that although the participant favors one side of the discussion, it may be willing to hear other parts.
Dishonest does convey both the hiding and the favoring of another side, but only if the participant was supposed to reveal his side at start.
I've stumbled upon a "concern troll" slang term that matches this meaning quite well.