I've been doing some research on the use of urban gang and mob violence as a political tool during the late Roman Republic (about 140–40) and was curious if there are any contemporary sources that justify or explain the violence. Some of the main figures to use urban violence for political ends include the Gracchi, Saturninus, Drusus, Catiline, and most famously Clodius and Milo (roughly in chronological order). From what I can tell, urban political violence was mostly carried out by populares senators but especially by the tribunes of the plebs. There are many texts (mostly Cicero) justifying the use of violence against these men, but I haven't come across any that justify the use of violence by them.
How utterly bizarre, I wrote my undergraduate thesis on more or less exactly this subject. Organized political violence, that is. So you may well be my new best friend. I'm sorry to say that pretty much every single scholarly treatment you're likely to have looked at is horribly outdated. Even Tatum's recent (and mostly pretty good) biography of Clodius, The Patrician Tribune, has already been rendered outdated by Bendlin's marvelous monograph on the Lanuvium Inscription, which overturns everything we thought we knew about collegia and whose findings have yet to be integrated into treatments of Clodian violence.
Organized political violence was hardly restricted to the tribunes or the populares (whatever we mean by that nebulous, more or less meaningless term), although you're right to suggest that often it was used as a response (I would still say not always, though). Part of the problem, after all, is what constitutes a popularis. Cicero, for example, often excludes people we might normally place under the label. For example, was Milo popularis? Was the elder Drusus, when he founded twelve colonies (a proposal typically associated with the urban plebs, who benefited from them more than agrarian laws)? Setting that question aside, there's little in the way of justification for political violence of any kind, except coming out of Cicero's mouth. Certainly Cicero is the only contemporary--the Gracchi, Saturninus (and Glaucia!), both the Drusi, Sulpicius, Clodius, Milo, etc. have not survived in their own words. This is part of what makes the study of this sort of thing so difficult, we have a severe lack of material and what we have is often either overtly hostile or much later (or both). But we wouldn't be doing this if the sources made it easy. In a very few cases our authors put words into the mouths of their characters or provide justifications for them. Sallust, for example, has Manlius claim that his open revolt (the Catilinarian violence is really a civil war, and it's noteworthy that no violence occurred in the city, which seems not to have been targeted by Catiline's legal program, if it existed) is in defense of libertas against the cruelty of the senatorial class and the moneylenders that drove Manlius' Italians into debt. Sallust even cites the secessiones plebis and justifies them by pointing to the arrogance of the magistrates and the desire by the plebs to exert the political power theoretically owed to them by the existence of the legislative assemblies (see Mouritsen, Plebs and Politics, to get a handle on the difficulty of that question). That seems like a decidedly Sallustian, post factum justification, the sort of thing you'd see in a treatise on political philosophy. Which is not necessarily enough to discount the basic idea. In another example, Appian notably absolves Ti. Gracchus of the initial show of violence, claiming that Octavius was not assaulted during the vote to abrogate his magistry. Plutarch, on the other hand, says that Ti. ordered his apparitores (who were also, apparently, his own freedmen) to pull Octavius down from the tribunal by force. Appian's account therefore is given a very different tone, in that Ti. throws the first punch only when he assaults the Capitoline, well after the threat against him has been prepared. Appian gives little overt justification of Ti.'s actions, but he makes it pretty clear that Ti. was responding to a real social problem (albeit one that is in fact highly debated now--was there a manpower crisis at all, or was it imagined?) and that he was faced with insurmountable, illogical opposition that could only be met with force. At the same time much of this is feeding into Appian's depiction of the period from around 133-80 or so as some sort of teleological thread leading inevitably to the Social War and the eventual enfranchisement of the Italians.
Another part of the problem, related to the insufficiency of our material but separate from it, is how we are to view these violent agitators. Cynical demagogues deploying their minions for their own benefit? Champions of the plebs selflessly fighting for what the plebs wanted with little thought for themselves? Neither view makes a lot of sense, and naturally these extreme views fail pretty consistently. Still, how did these individuals present themselves to their violent followers? Who were their violent followers to begin with? My thesis spend nearly a hundred pages barely coming to any conclusion about Clodius, let alone his predecessors and successors, but we're presented with some realities. As was identified at least as early as Lintott, many of these individuals are expressly described as leading a core group of their own clients and friends--both the Gracchi appear to have had a core group of three to four thousand. The role of clientela in Roman society and politics is a subject under concerted frontal assault these days, with good reason, but it remains pretty undeniably true that, given what we can glean from the texts, these people often deployed relatively small, personally loyal groups--mind you, such generalizations hardly apply to every individual politician over the course of over a century (but I don't have nearly a hundred pages on reddit...). At the same time, the population at large could and did become involved often. When Opimius was empowered by SCU against C. Gracchus Appian tells us that most ("most") of the urban plebs assembled in the forum to await his orders. This view of relatively small groups, often containing a large proportion of personally-associated individuals appears to align pretty neatly with what we know about oratory and "popular" politics in general--both Mouritsen and Morstein-Marx have made similar arguments with respect to contiones. At the same, violent politicians often appear to have paired their violence with some sort of idea of necessary legislation (for what group is another question) that had to go through in the face of overwhelming political resistance. While we cannot know that violent politicians often depicted their actions as extralegal means to work around what was, as Mouritsen has ably shown, really a rather restrictive political process, the circumstances and their actual actions often suggest this. For example, a classic, arguably the classic, tactic of the violent tribune was to disperse the comitia by force, either preventing a vote or securing the comitia and setting up a unilateral vote. Why any given politician's supporters were of the opinion that this particular vote had to go through is not always clear--why on earth was it so important to transfer the Mithridatic command to Marius, for example? And the possibility of essentially mercenary muscle cannot be discounted. The use of gladiators in personal retinues (who, I argue, pretty much always made up the core of a politician's violent forces) is well-attested, and we have references from, say, Sallust regarding what appear to be some kind of essentially violent bands-for-hire (duces multitudinum).^^1 These forces, which might have formed the core of any violent group, were not necessarily so idealistic. Yet we have evidence that normal people often joined in, and it's not always clear why. Often we find underrepresented groups participating in violence. Appian says the Italians often did in the run-up to the Social War, fighting against the urban tribes whose already minor influence in the Tribal Assembly was further diluted by the eventual enfranchisement of the Italians.^^2 Towards the end of his life Clodius appears to have made an overt gesture towards Roman freedmen, who were confined to the four urban tribes and therefore only four votes, an effectively useless margin of power. Would this have served as a justification, that since it was unlikely for the distribution of freedmen into all the tribes to pass in the face of pretty much any opposition at all the only recourse was to push forward what seemed a necessary legislation to some by force? Hard to say--even sources like Cicero don't seem to make this sort of connection. "Hard to say," in fact, generally sums up this whole problem. We really don't know. And it's a question not just confined to the arena of political violence--how these sorts of actions were depicted by those committing them is a problem that encompasses the entire ideology of the Republican state.
The question of Clodius' operae is a related, but extremely fraught problem that I don't have the time or space to deal with here. Let it suffice to say that I see very little reason to believe the "traditional" view of the term operae that can be found in scholarship going back to Mommsen. It's grounded on flimsy evidence, bad readings of Cicero, and a great deal of speculation about the composition of various social associations, many of which were straight up invented by Mommsen (Bendlin's monograph is priceless in breaking up these myths). In fact I can find very little evidence that Clodius' forces were especially different from those of violent politicians before him, at least in the way they were organized and deployed.
It's also worth noting that prior to Clodius, or at least about 60 or so, I have been able to find startlingly little evidence for violent tribunes in our texts championing urban proposals first and foremost. Why this is, and whether it's a perception bias given from the point of view of our sources, is a difficult question.