Some X-ray bursts from sources such as 4U/MXB 1636 - 53, 4U 1608 - 52, and possibly MXB 1727 - 34, display a multipeaked structure in the bolometric flux that is difficult to reconcile with the standard thermonuclear flash model or with the effects due to photospheric expansion and contraction. Understanding this phenomenon is important primarily because it might be due to a variable Thomson optical depth along a line of sight threading the burst-induced accretion disk corona. We explore the conditions under which these burst profiles might be observed, in light of the somewhat surprising recent results from the radiative-hydrodynamic simulation of the coronal structure. We find that double- (and triple-) peaked bursts are produced at inclination angles in the range 40-65 deg. Interestingly, the orbital information for 4U/MXB 1636 - 53 and 4U 1608 - 52 suggests i is less than 70 deg, in these sources. The orbital inclination of MXB 1728 - 34 is not yet known.