Hemianopia patients have lost vision from the contralateral hemifield, but make behavioural adjustments to compensate for this field loss. As a result, their visual performance and behaviour contrast with those of hemineglect patients who fail to attend to objects contralateral to their lesion. These conditions differ in their ocular fixations and perceptual judgments. During visual search, hemianopic patients make more fixations in contralesional space while hemineglect patients make fewer. During line bisection, hemianopic patients fixate the contralesional line segment more and make a small contralesional bisection error, while hemineglect patients make few contralesional fixations and a larger ipsilesional bisection error. Hence, there is an attentional failure for contralesional space in hemineglect but a compensatory adaptation to attend more to the blind side in hemianopia. A challenge for models of visual attentional processes is to show how compensation is achieved in hemianopia, and why such processes are hindered or inaccessible in hemineglect. We used a neurophysiology-derived computational model to examine possible cortical compensatory processes in simulated hemianopia from a V1 lesion and compared results with those obtained with the same processes under conditions of simulated hemineglect from a parietal lesion. A spatial compensatory bias to increase attention contralesionally replicated hemianopic scanning patterns during visual search but not during line bisection. To reproduce the latter required a second process, an extrastriate lateral connectivity facilitating form completion into the blind field: this allowed accurate placement of fixations on contralesional stimuli and reproduced fixation patterns and the contralesional bisection error of hemianopia. Neither of these two cortical compensatory processes was effective in ameliorating the ipsilesional bias in the hemineglect model. Our results replicate normal and pathological patterns of visual scanning, line bisection, and differences between hemianopia and hemineglect, and may explain why compensatory processes that counter the effects of hemianopia are ineffective in hemineglect.