It seems like the translation matrices for the BHI160 aren't completely consistent. Using the "BHI160" app to read the sensor data, it looks like the accelerometer is a left-handed coordinate system. If the badge is lying flat on a table:

• X points right
• Y points forward
• Z points down (into the table)

However the gyro axes aren't the same (if we assume positive values being clockwise rotation around the directional vector):

• X points right
• Y points forward
• Z points up

As can be seen, the Z axis is inverted. This means that while downward acceleration (like gravity) causes positive accel values, clockwise rotation around the vertical axis causes negative gyro values.

To try to fix this, I changed the translation matrix in bhi.c to invert the Z axis:

-static int8_t bhi160_mapping_matrix[3 * 3] = { 0, -1, 0, 1, 0, 0, 0, 0, 1 };
+static int8_t bhi160_mapping_matrix[3 * 3] = { 0, -1, 0, 1, 0, 0, 0, 0, -1 };

This does indeed fix the gyro axis! Gyro Z now points down, with clockwise rotation causing positive values. This matrix is used for both the gyro and the accelerometer though, so the accelerometer should have been inverted too - which doesn't happen. In fact the accelerometer seems to be completely immune to this matrix, which may be a separate issue.