Yes, they are independent measurements, even if loosely associated within the overall navigation system. Strictly speaking, the IMU is only the Inertial Measurement Unit, which is essentially a set of three orthogonal solid state gyros that fundamentally just measure acceleration in each of the three axes. Those acceleration measurements can be integrated in software to obtain velocities in three dimensions, and those velocities can be further integrated to obtain position in three dimensions (which is also corrected with the slower GPS data, to control gyro drift). You'll notice in your IMU calibration data that it always ends up with zero acceleration in the x-y plane, and -1 acceleration in the z plane (for the acceleration of gravity) which are all correct for the rest state used in calibration. If you get two zeros and a -1, for x, y, and z respectively, you have completed a good IMU calibration.
The barometer is simply a pressure sensor only, not oriented with any spatial dimension. It's just an altimeter, except it isn't calibrated for local barometric pressure as would be needed for absolute altitude; it just measures relative altitudes.
Likewise, the compass is an independent sensor that measures magnetic field strength (in two dimensions) for heading reference and spatial orientation. The flight computer puts all this disparate sensor information together with software to support the overall system navigation, and for information display to the user.