In liquid crystal pixel cells, it is only the magnitude of the applied voltage which determines the light transmission (the transmission vs. voltage function is symmetrical about 0V). To prevent polarisation (and rapid permanent damage) of the liquid crystal material, the polarity of the cell voltage is reversed on alternate video frames. Unfortunately it is very difficult to get exactly the same voltage on the cell in both polarities, so the pixel-cell brightness will tend to flicker to some extent at half the frame-rate. If the polarity of the whole screen were inverted at once then the flicker would be highly objectionable. Instead, it is usual to have the polarity of nearby pixels in anti-phase, thus cancelling out the flicker over areas of any significant size. In this way the flicker can be made imperceptible for most "natural" images.
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The inversion pattern for any given screen will inevitably flicker to some extent and is not a fault. If it really flickers a great deal then it may indicate that the common-electrode voltage has not been set up properly. In that case you might also perceive a "dot crawl" effect on plain colours of medium brightness. A grossly mis-set common-voltage will also make your screen more susceptible to temporary 'image sticking' problems.
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Laptop LCD screens tend to be optimised for lower power, with some relaxation of image-quality criterion. As well as (often) lower brightness and less saturated colours, laptop screens usually use a 'row inversion' (aka 'line inversion') scheme rather than the dot inversion now universal in desktop screens. If you look closely at a row-inversion LCD, particularly if it is showing a fairly plain, mid-brightness colour, you may see a slight horizontal line interference pattern on alternate lines, which may appear to drift up or down the screen. This is also not uncommon on colour mobile-phone displays, or personal DVD players.