Yes, there are currency pairs that move the same direction. They are called correlated pairs. The level of correlation determines the similarity in the currencies' moves.
There are pairs that move in the same direction almost mirror-like:
EUR/USD and GBP/USD
AUD/USD and EUR/USD
And pairs that move mirror-like, but in opposite directions:
EUR/USD and USD/CHF
GBP/USD and USD/JPY
By knowing which currencies have the highest chances to reflect each other moves traders can diversify the portfolio, instead of putting all eggs in one basket.
Further reading and examples of other currency pairs at:
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