Rivalries in sports come in two basic flavors. There are rivalries borne of repeated competitive meetings of great import and there are geographic rivalries.
The Los Angeles Clippers have participated in a grand total of 11 playoff series in the franchise's 42 years of existence, and four of those occurred in Buffalo in the 70s. (I'm not doing the research on this, but I feel fairly safe in stating that 11 playoff series is probably the fewest of any NBA franchise that has been around 42 seasons.) In those 11 series, they've faced nine different teams, with the old Buffalo Braves losing on two different occasions to the Boston Celtics in the 70s and the Clippers losing twice to the Utah Jazz twice in the 90s. The other seven were playoff opponents only once, which then leaves 20 other NBA teams the franchise has never faced in the playoffs. Despite 82 of them every season (or almost every season), the simple fact is that regular season games are rarely overly important. Playoff games are where competitive rivalries germinate, and the Clippers simply don't have much soil for planting there.
On the other hand, few teams in the history of sport have a more obvious geographic rival. After all, how many sports teams share an arena?
For the Clippers, there's only one rivalry, and that is the Lakers. Of course, there are probably 29 other NBA teams who consider the Lakers their rival, and about 20 of those may be higher on the Laker radar than the Clippers -- or at least they were until recently.
Even when rivalries are geographic in nature, competition is still a significant factor. Westchester is closer to Westwood than downtown L.A. is, but it doesn't mean that Loyola Marymount is a bigger rival for UCLA than USC. In the first 25 seasons after the Clippers moved to Los Angeles, their record against the Lakers was 25 wins and 90 losses. The Clippers have never won a Pacific Division title and have finished ahead of the Lakers in the standings on just four occasions. Only twice have the Clippers been further in the post season than the Lakers.
For all of those reasons, it's understandable if the Lakers have not considered the Clippers much of a rival over the years. However, in recent meetings the games have taken on that extra level of urgency that is frequently associated with rivalries. From Blake Griffin and then-Laker Lamar Odom getting into a shoving match, to Griffin posterizing Pau Gasol not once but twice, to Gasol ticking Chris Paul off by patting him on the head, the games of the last few years have been intense.
Not to mention the fact that the Clippers are currently the primary competition for the Lakers in the Pacific Division, and a factor in the entire Western Conference as well. The Lakers finished a scant game ahead of the Clippers in the standings last season and it figures to be close again this season.
As long as Blake Griffin and Chris Paul are playing for the Clippers, the Lakers will have to deal with a new paradigm in the Hallway Series. The Clippers are a very dangerous, and very high visibility team now and for the foreseeable future, which could make for a very good rivalry indeed going forward.
When it comes to these two teams, there has been no balance in the city itself over the years. For every Clipper fan in L.A. there are at least 20 Lakers fans. Clearly this is one aspect of the rivalry, not only for the fans (Clipper fans get very tired of being in the minority and being put down) but also for the players. Paul and Griffin are highly competitive, and they would like nothing more than to make inroads into Laker territory. They'd like to walk into a store and see Clipper gear for sale next to the Laker gear. For their part, the Lakers still see plenty more purple and gold T-shirts and car flags around town, but they also see Clipper coverage on Sports Center night after night, and they know that they're no longer the only marquee team in town.
For now at least, the rivalry is real, if a little one-sided. The Lakers have many traditional rivals, but there is one and only rivalry for the Clippers. In a poll on Clips Nation, when the Lakers were taken out of the equation, only the Grizzlies and Thunder got more votes than "None of the above" among Western Conference teams. The Memphis votes are clearly skewed by the fresh memory of a hard fought and somewhat nasty playoff series a mere four months ago, and as of now any rivalry with Oklahoma City is more wishful thinking than anything else (though one can indeed imagine a near-future Western Conference where Westbrook and Durant are squaring off against Paul and Griffin in the playoffs for many years to come). As for any potential Eastern Conference rivals, all 15 East teams finished far behind "None of the above". There are teams Clipper fans don't like (hello, Miami), but no rivals.
So it's all about the Lakers as far as the Clippers and rivalries are concerned. And that's as it should be. The Lakers are arguably the most successful franchise in the NBA and they share a city and an arena with the Clippers. What more could you ask for in a rival? It's up to the Clippers to make it competitive now.