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In terms of how interested you are in the comparison, how heated they seem, how comparable they seem, how much you personally are invested in comparing the two:
I would say the strongest ones are probably WITHIN countries, often the two biggest cities. In many cases it's the capital vs the biggest city when the capital is not the biggest city. Typically the biggest city became that way by being the main commercial centre.
Beijing vs Shanghai
Delhi vs Mumbai
Hanoi vs Ho Chi Minh City (Saigon)
Would be the examples of this city rivalry.
Also:
Hong Kong vs Shanghai
There are also rivalries within countries where one city is much more prominent but where the other city, perhaps due to an inferiority complex, still likes to compare themselves to the alpha anyway:
Tokyo vs Osaka being the best example of this, or Taipei vs Kaohsiung
Between countries, there are many rivalries too. These are often between cities of comparable size/important, although in some cases this might extend to food/shopping/tourism like Singapore vs Kuala Lumpur:
Hong Kong vs Taipei
Singapore vs Kuala Lumpur
Singapore vs Hong Kong
Tokyo vs Seoul
Tokyo vs Shanghai
Shanghai vs Seoul
Kuala Lumpur vs Bangkok
Singapore vs Bangkok
Dubai vs Mumbai
Dubai vs Hong Kong
Istanbul vs Tehran
Then there are smaller cities. One that is a potential comparison/rivalry is:
Kyoto vs Suzhou
I guess for me Singapore vs Kuala Lumpur since I'm most familiar with the two. They're pretty different to compare, each with their own strengths, but I'm biased. I find Hong Kong vs Shanghai an intriguing comparison as well. Delhi vs Mumbai looks like quite a even comparison when you throw in tourism and government for Delhi vs Bollywood and IT wealth in Mumbai.