There are plenty of Hindus who reject the caste system. So it can’t be integral. Indeed, the only thing I can think of that is integral to Hinduism is its Indianness
Speaking as a Hindu, this is my view of Hindu-ness. You don’t have to believe any particular thing to be a Hindu. You don’t have to engage in any particular practice to be a Hindu.
What is integral to Hindu-ness in my view? You become a Hindu by being born a Hindu. There’s nothing you can do or fail to do that can disqualify you from being a Hindu.
You can add Guru Nanak or Muhammad or Jesus Christ to your prayer table. You can subtract any Hindu god. You can subtract all gods. You can be an atheist. But if you believe in any one deity, you cannot declare that any other deity is false.
You can be omnivorous. You can eat everything except beef. You can be an ovolactovegetarian. You can be an “Indian” vegetarian. You can be a “pure” vegetarian. You can be a vegan. No type of diet will disqualify you from calling yourself a Hindu.
You can believe that your caste is important and that other people’s castes are important. You can believe in caste and yet believe that no one caste is better than another. You can deny caste and declare the caste system to be a form of discrimination, bigotry, and oppression.
Not that it matters, but another Hindu (atheist) who agrees fully with you there.
Hmmm…isn’t this a potentially problematic issue in a few areas of India like Kerala?
Many Hindus I know do eat beef. Originally it was not a religion thing, but over time it has become.
The way it was explained to me : Cows were used to plough the fields in India much like Horses were used in Europe.
People did not eat the animals that ran the farm because every few years, the monsoon will fail in India resulting in famine. If humans ate the animals that ran the farm during famine, then when the rains came back ; there will be no animals to plough the field. Which in turn will result in longer and bigger famines.
It takes a lot of conviction to not eat the cows when you and your family is starving during a famine. And that societal rule has become something intrinsic to Hinduism because before modern irrigation, the country lived on monsoons.
Hindus did eat horse meat in legend / folklore maybe because unlike Europe, Horses were not used to plough fields in India. A practice among the kings was “Ashwameda yagya”. Basically a king will let loose a royal horse and soldiers will follow the horse wherever it went. If it went to another King’s territory, the other king had the option of capturing the horse and call a war or surrender to the horse owner. This culminated with the horse being slaughtered and consumed when it was all over.
What do you mean?
All this is also my understanding. I’m fact many religious practices originated in environmental conditions, including the avoidance of pork in the Middle East.
This is especially true of Hindu practices, most of which arose over time and aren’t grounded in scriptures or anything like that.
My understanding (which I’m happy to be corrected on) is that whether through acculturation during the colonial period or earlier cosmopolitan influence from overseas trade a fair number of Malayali Hindus have taken to eating beef. Hence its continued legality in Kerala - i.e. it is not strictly limited to the Christian and Muslim populations in that state.
Then there are the question of Dalits, some of whom I also understood to eat beef.
ETA: To more explicit what I mean is isn’t a definition of Hindus that excluded people who ate beef potentially problematic given there some parts of India where Hindus do eat beef, even if not heavily.
Kerala by virtue of being a coastal state and easy to access by sea has had many influences over history. Jews took refuge in Kerala during the times of King Solomon circa 950BC. The first mosque in the subcontinent was built circa 650 AD by a companion of the Prophet Mohammad.
Their food must have been similarly influenced. I can totally see them eating beef and have no “beef” with it.
Food is influenced more by geography. In a state called Assam, in India (maybe you know it from tea), some people eat dog meat. Some Hindus believe only vegetarian food be consumed during festivals ; others insist on having meat.
Even among some Shia Muslim factions , they consider beef as Haram in India.
The point is that food is evolving all over the world, there are some religious zealots in every country who control the narrative around what’s food, but hopefully the zealots will not win.
As shown in the conversation above, diversity, variety, and contradiction are inherent to Hinduism. But avoidance of beef eating is a widely accepted definition. It’s becoming politicized now and is becoming a huge problem, not just in Kerala.
So…it’s Nationalism?
Some people have made it into nationalism. The current political climate speaks to that.
Right, but didn’t you just say that?
Yeah, you guys can take it to the next level. 
I think there are two separate issues being conflated here. You seem to be referring to who has the status of “Hindu”. That’s distinct from who is a believer in Hinduism. (The same is also true of Judaism vs being a Jew, where a person who is born Jewish has the status of Jew per the Jewish religion regardless of what they personally believe.)
So if when you say you can reject the caste system and still “be a Hindu” you mean it in the context of the assertion that you can reject any Hindu religious tenets and still “be a Hindu”, then it’s not meaningful for the purpose of this issue.
What’s at issue here is the extent to which someone can be a believer in the core tenets associated with the Hindu religion (at least in commonly accepted form) and still reject the caste system. Because if the only way to reject the caste system is to make wholesale revisions to (or reject outright) significant other portions of commonly accepted Hindu religious beliefs then that’s going to be a tough sell.
There is no such distinction to be made, Hindu versus Hinduism. Hindus as they exist define Hinduism. Hinduism is anything that a Hindu does. Hinduism has no dogma. There are no necessary or sufficient set of beliefs.
In that view, perhaps the only widespread Hindu attitude is that it’s nigh impossible to become a Hindu if you aren’t one already.
Hinduism is unlike any other major religion in This sense. That’s why there are people who say things like “Hinduism is not really a religion.”
No, I don’t think so. You might say that it has aspects of “being a nation,” but not necessarily of “nationalism.”
I’d be skeptical of any attempt to assert that any tenets are core. There are more-or-less common elements of belief, but no one of them is necessary.
The Indian constitution defines as a Hindu any non-tribal Indian who is not a Christian, Parsi, Muslim, or Jew.
So, the legal definition of Hindu is not based on any list of tenets or beliefs.
Interesting. Does that mean henotheism is a required dogma in Hinduism? I see the wikipedia page on it. Can you explain more?
That was part of my personal view of Hinduism.
There is no required dogma in Hinduism. There are monotheistic Hindus. There are anti-Muslim Hindus.
And for everything I say about Hinduism, someone else exists who will contradict it.
It can’t be pinned down.