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Is there a relationship between tensor (or form) bundles and iterated tangent/cotangent bund...

The answer is 'basically, no'. The tensor bundles that you list, such as $T^{(n,m)}M$ and its quotients (such as $\Lambda^p(TM)$, etc.), are first order prolongations of $\mathrm{Diff}(M)$, whereas $ …
Robert Bryant's user avatar