Osaka vs Tokyo
Being a major trading and industrial city in Japan, Osaka had to allways compete with Toyko. Once Tokyo replaced Kyoto as Japan's administrative center and then became the capital, Osaka could not keep up in power and influence. Also after World War 2 companies mostly established their many offices in Tokyo, making it even harder for Osaka to keep up. That said Osaka still conducts major trade with Tokyo, which is perhaps its highest trading partner, since the megacities are relatively close by.
Other Cities
The city and prefecture of Osaka has friendship ties-ups/close relations and agreements with 10 cities in 9 different countries, as with the megacity of Shanghai in China. In these places the Prefecture offers business support, local reports, business transactions, and supports business trips, along with conducting trade with the certain place.