The core of the subject of whether or not there is a sect called "khawarij" depends entirely on your technical description of the word. Muslim scholars use different expressions to imply the technical meaning of the term "khawarij" but a technical description MUST, absolutely MUST contain who and what the Khawarij are to be concrete.
Al-Shahrastani [Al-Milalu Wa Al-Nihal Vol. 1, p. 129] has defined the Khawarij as: "Anyone that goes against a right leader upon which the majority has agreed [to have him for a leader], is called a Khawarij; be it against the Rightly-Guided Caliphs during the time of the Prophet's companions or against the followers of the Prophet's companions as well as against leaders at any time."
Al-Shahrastani's classification of course makes imperative the defining also of a "right leader".
Shia believe the right leaders of Islam to have been the Prophet Mohamed's family, daughter Fatimah, and son-in-law Ali due to interpretation of hadith, and following, their children. Sunni believe the companion Abu Bakr, the prophet Mohamed's father-in-law due to his marriage to Aisha, the same, according to the same hadith by another interpretation and consensus at the time of Rasoolulah's death, up until Ali, and sometimes Umar ibn Abdul-Aziz.
I personally believe the same definition of "rightly guided" as Al-Sharastani. That any Caliph [rightly guided leader of the Muslims] is just any pious and just Muslim leader that leads a Muslim state in compliance with Islamic law, as the Prophet Mohamed salalalhu alahi wa salaam did. Though for reason that will be explained shortly, I'm not sure if I buy his technical definition yet.
I believe the first Caliph would have to have been
a.) Appointed in accordance with the teachings of the Holy Qu'ran and Prophetic traditions. [I favor Abu Bakr, so do Sunni, and Ibadhi though both do not agree on the historical leadership after Umar, and Shia favor Ali]
And any Caliph existing now or before would have/had to:
b.) The Caliph must lead in accordance with those teachings and does not go contrary to them.
c.) The Caliph has not stepped down nor has he been deposed.
Al-Shahrastani means by rightly guided Caliphs specifically: Abu Bakr, Umar, Uthman, and Ali. If that is his technical definition as intended, then many Al-Shahrastani would never have labelled as Khawarij also fall under that label, including Talha, Zubair, and Muawiya... who rebelled against Ali [rightly guided Caliph] during his Caliphate.
That is the flaw in Al-Shahrastani's [and most Sunni scholars who I have thus far examined] classical technical definition I suppose [though I'd put modern day Qutbists and Osama Bin Ladens as Khawarij with no problem in breath and no Ibadhi would stop me from that] . That this technical definition's "what" does not make clear its "who".
Al-Shahrastani's definition of "what" the Khawarij are may encompass Aicha R.A, Talha, Zubair, and Muawiya, but when you ask the scholars of both Sunni and Shia who the first Khawarij were they will say the people of Nahrawaan (the Ibadhis).
But the funny thing about Sunnis especially holding the people of Nahrawaan as the first Khawarij not Talha and Zubair and Muawiya? Well, for one, Ali was no longer Caliph in the Sunni take on history when the people of Nahrawaan "went out" from his leadership.
Another meaning of "a right leader" can also be taken to mean a Muslim leader regardless of the means by which they have come to power or the legitimacy and legality of his political existence according to Islamic law.
I don't agree with a "Caliph" ever being this.
If this definition of a right leader is used then I am for sure Khawarij and proud of it. But historically, so would ever Muslim in history be. Sunnis would be Khawarij because they sided with Muawiya, Ibadhis would be Khawarij because they disagreed with Ali, Shia would be khawarij because Al-Hasan ibn Ali revolted against Muawiya, and Al Hussain organized a campaign in Iraq against the Umayyad leaders.
That doesn't work to clarify at all, does it.
Thus we can conclude that we are either lacking a a more definitive definition of "right ruler" or that Al-Shahrastani's "what" definition of Khawarij, is, er, lacking the ability to clarify the "who".
When trying to research this technical definition I ran into a problem. The technical definition is supposed to be the 'what' and the 'why'. Most scholars ignore what "khawarij" is supposed to be, assuming already that the people of Nahrawaan are "it" whatever that is disregarding the flaws of trying to be technical about it all, and launch into merely giving reasons they have been thus labelled, and listing actions they are purported to have done/do.
Most commonly, as related by Ibn Hajar [Al-Fat-Hu, Vol. 14, p. 8524] and Al Shawkani [Sharhu Al-Nayl, Vol. 7, p 164] we hear: "The word Khawarij is a plural form of the word Kharijah, which means group. They have been so-called because they have gone out of the religion and because they have innovated matters once not existed."
Ibadhi will argue Sunni innovate, Shia will argue Sunni innovate, Sunni will argue Ibadhi and Shia innovate. So that definiton (technically unsound as it is lacking "what" and "who") will be no avail to this study either.
One problem I have found in my study so far of the Khawarij (and please enlighten me if you found a better what and who then I have) is that in both classical and modern works on the subject there has been no cohesion between the definition and the conclusion. And that often when a definition of "what" is used, other that fall under "who" of that what are excluded for no logical or scientific reason.
The most respected statement (of course not by the Khawarij themselves;D) about the Khawarij is by Ibn Taymiyyah Al-Harrani [Al Tasfir Al Kabir Vol. 2, p. 8]. Ibn Taymiyyah says as for "who": "Khawarij are the first callers of disunity in Islam." Explaining: "The first disunity in Islam [was] after the murder of Uthman and the division of the Muslims, and when Ali and Muawiya agreed on making arbitration, the Khawarij opposed and proclaimed that: 'There is no judgement except that of Allah' and they disassociated themselves from the mainstream."
The Shia were already here, divided from the Sunni, and likewise, so any student of history might wonder at Ibn Taymiyyah calling the people of Nahrawaan [who historically opposed the arbitration between Muawiyia and Ali as a trick of Muawiyia's] the first callers of disunity. I know I do. Another case of same cases, different rulings. Also, Ibn Taymiyyah's definition of splintering one's self from Ali alone would not make one Khawarij as Ibn Taymiyyah himself does not consider companion Ibn Abbaas who left Ali (who personally considered war agaainst those of Nahrawaan unjustified) for Mecca, to be Khawarij. Even if you changed his definition to mean " the act of splintering from Ali combined with war against Ali" you'd have to say that Talha and Zubair were Khawarij too, and no one does. Some use the term "disbelievers" but not specifically "Khawarij". What's the difference one might ask? To go out from Islam is to disbelieve.
More to come. What do you think so far? Who are the Khawarij to you and if so, do you know why you consider them Khawarij? What does one have to believe/do to be Khawarij?