^EUU price means 1 Euro = $1.3741 (so EUU down means USD up). ^EUI price means $1 = .7274 Euro (so EUI up means dollar up). Neither of these have an underlying so they go off of the spot rates.
The Charts Tab snapshots for each are included (^EUI and ^EUU respectively) (click either image to enlarge).
Note how the dollar has strengthened against the Euro of late (^EUI up trend or ^EUU down trend). Note also the IV30™ (red line) increasing in the bottom half of the charts (the vol charts) vs the HV20™ (blue line).
So an interesting question is what the market is implying moving forward. The Skew charts for both are included (click either to enlarge).
Currency skews are not like stock skews. Currencies usually have parabolic skews - not a smirk. You can see pretty clearly in the skew charts that ^EUU has a pronounced downside skew in the front months (red, yellow, green). The three outer months (light blue, dark blue, purple) are flattish parabolas ("normal" currency skew). This indicates that the near term options are priced for downside vol (downside move) - that is an upside move in the USD relative to the Euro.
The ^EUI skew has the same result with the opposite picture. The front months (red, yellow, green) are skewed to the upside, while the back three months (light blue, dark blue, purple) are more flat parabolic (no lean).
The currency options are pricing greater risk that the dollar goes higher versus the Euro.
This is trade analysis, not recommendations.
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