There's a good bit about this in Lundy - I don't have my copy any more but will try to remember.
From memory/extrapolated experience, the abuser will tend to come across very sincerely in the counselling, perhaps even convincing the counsellor to take his/her side. This is especially so if they are quite a charming person generally, and/or very respectable/come across as a pillar of the community type.
Then later, they take everything YOU've said and use it against you, when you thought you were in a safe space. They haven't actually said anything to incriminate themselves, and may spend quite a lot of time navel gazing about their own "issues" which the counsellor will see as a good thing because this means they understand why they act as they act and could choose to change (the chilling reality of this, of course, being that they know what they need to do to change but they choose not to).
Counselling tends to work on a premise that there are communication issues - not so in an abusive relationship, the abuser can hear you communicating your needs, they just don't care about those needs. You can hear what they are communicating quite clearly too, it's just that what they are communicating is highly unreasonable. (e.g. "You should never go out with other men in case they hit on you"). Abusive relationships are sometimes mistaken for communication issues, because the victim assumes that the abuser is not hearing or understanding their needs, rather than dismissing them. Also, the notion that the abuser might mean what their messages are telling you (the unreasonable expectations) is very difficult to believe, especially if you essentially believe that your partner is on the same page as you. That's the problem; with abuse they are often not. (And they believe that their page is far more relevant and valid than yours).
Or it works on a premise that both partners are equally flawed and need to work as much as each other to fix things. This is, again, not the case where abuse is present, because it is the ABUSER who needs to make changes, and they have to be very drastic changes, not the usual "well it's improving..." that might be acceptable with other communication issues. It is not possible for an abuse victim to simply "stand up to" an abuser, in that it doesn't work and usually just enrages the abuser more/looks like an increased challenge. The abuser genuinely believes in a difference of power or status in the relationship, much like parent and child or master and dog. They do not expect to be "stood up to" and if it happens it is a sign (to them) that they are dangerously out of control and drastic measures must be taken.
The abuser believes they are justified and will come across as such. I don't know if you saw the Channel 4 doc on adoption the other night but there was a very abusive young man on there who described the reason his child was removed as "arguments and stuff". Later in the programme it was shown that the police had been called out to the couple on several occasions, so clearly not most people's idea of normal arguments. An abusive person's perception of concepts like "arguments" "shouting" "violence" and also "love" and "support" might look very different to an emotionally healthy person's understanding. Similarly, victims are often self-deprecating and willing to accept blame, almost exaggeratedly so. "Well it wasn't really his fault, I know how easily he gets wound up and I kept going on about it, I couldn't just leave it. Stupid." They get caught up in the abuser's reasoning and justification of their actions and in some cases start to believe they are genuinely at fault. Because it's taken for granted (on both sides) that this is the case it may not be obvious to an outsider that the interaction is skewed, especially if they are being nonspecific about incidents.
There is probably more, but the short version is that abusive characteristics can't be solved by talking it out in counselling, they need really intensive therapy which itself has a very low success rate, and often is impossible to resolve within a relationship because the history is too great to overcome.