Not calendar spreads. But, intermarket index spreads. Calendar spreads on index futures are sensitive to interest rates. But, bond spreads also move the index because of no arbitrage pricing. The pricing model involves a floating rate. (risk free rate)
The pro's will trade the intraday performance using hedge ratios. They're getting exposure to the actual % change in value of the stocks comprising the indexes.
(long/short trading in the front month contracts)
It's a constant battle for liquidity, since there are so many shops using the index as a hedge in one way or another.
YM/ES is heavily traded, as is NQ/ES due to overlap in the constituents.
On Friday, the YM/ES spread had a massive rally right at the open. The move was huge because the dow had sold off >1800 points the day before.
It was short covering. Just think about it. When the NYSE opening bell starts ringing, and all the orders hit the cash market, each index is gonna jump relative to the others. It's not really possible for every index to move in a completely controlled way.
Futures pro's are trading the divergence using VWAP and technicals. There are also ways to lever index spreads using structured products. (return swaps)