Older (civilian) systems had gyro stabilised platforms with accelerometers mounted on a stable platform - that is, they remained in a horizontal plane. The popular Carousel system did complicate matters somewhat, by rotating the platform at 1 r.p.m. to enable any platform leveling inaccuracy to be isolated as a one cycle per minute error signal in the accelerometer outputs. However, all gyro stabilised systems isolated the accelerometers from the aircraft roll, pitch and yaw movements and held them in a level plane.
In a ring laser gyro system, accelerometers are fixed to the aircraft, aligned with the aircraft's roll, pitch and yaw axes. The laser "gyros" are used to accurately measure the roll pitch and yaw movements of the aircraft to enable the spacial alignment of the accelerometers to be determined by the processor. The resolution of the flight path is performed by computing the various vectors obtained from each accelerometer - each of which will simultaneously be measuring an acceleration in each of the three planes. The IRS computer resolves the complex pattern of accelerometer outputs into horizontal and vertical components by reference to the measured attitude of the aircraft and thus of the accelerometers themselves.
This is what I intended to convey by the term "strap-down" - that the accelerometers are not isolated from the aircraft attitude by a stabilised platform, but that the entire reference unit - including both the accelerometers and the ring lasers - is firmly attached to the aircraft with no moving parts.
The IRS measures the movement of the aircraft away from its starting point and in general cannot be realigned in flight. Indeed, for most systems in use today, the aircraft cannot be moved during IRS alignment on the ground and even aircraft loading can disrupt the alignment process. Position updates are performed by adding an offset to the computed position, the IRS continues to use its own computed position, then adds the offset to its output before displaying position to the crew and other aircraft systems.
You can find the value of this correction at the end of any flight by perfoming a terminal error check as per the AMM.