I accidentally and absentmindedly washed my hands for the meal on Shabbos as I would wash in the morning upon awakening, with switching off hands in between each pouring, pouring one time on the right hand and then one time on the left-hand, and so on and so forth three washings. Did I fulfill my obligation and should I have re-washed my hands properly?
So long as there is no worry that the water of the first washing of either hand got onto the opposite hand [which is guaranteed if you used a towel for washing as is the Chabad custom] then the washing is valid, and there is no need for you to wash again. [Nonetheless, initially we are particular upon washing for bread to wash each hand three times in a row, unlike the morning washing upon awakening, in which on the contrary, we are particular to switch off hands between each pouring.] Furthermore, even if there is a chance that your hand received the water from the other hand, if you are particular to pour 86 mL of water on each hand by each of the first pouring’s, then according to most authorities, there is nothing to worry about, although according to Admur in his Seder, it is proper for you to rewash without a blessing after properly drying your hands. If you did not pour 86 mL of water on each hand by each of the first pourings, then if there is chance that your hand touched the water of the other hand, then according to all opinions you should rewash without a blessing, after properly drying your hands.
From the letter of the law, there is no intrinsic obligation to wash hands for bread differently than in the morning and one can technically fulfill his obligation upon washing for bread in either fashion, whether one washes each hand three times in a row or switches off hands between each pouring as he washes in the morning. Nonetheless, despite this technical letter of the law allowance, the tradition of world Jewry is that upon washing for bread we are particular to wash each hand two or three times in a row [depending on custom as some only wash two times while others wash three times, although whatever the case everyone performs both pourings consecutively in a row]. This is unlike the morning washing in which it explicitly states in many Poskim [based on Kabbalah] to be particular to wash in a way that one switches hands between each of the three pourings in order to get rid of the impurity. Interestingly, we do not find any source in the codifiers for this contrasted custom that upon washing for bread one is to always be particular to pour all three pourings consecutively on each hand. Nonetheless, despite this being the case, there is a great reason behind why when washing for bread one should be particular to wash each hand consecutively, and that is in order to prevent one hand from touching the water of another hand prior to its second pouring. Indeed, this reason is recorded in the Poskim.
To explain: The law is that the first pouring that one pours onto the hand becomes impure if it contains less than a Revius of water [and according to the Raavad, even if it contains a Revius of water, and so suspects Admur in his Seder Netilas Yadayim]. Now, the law also is that if the impure water of one hand gets onto the other hand, then that hand must be completely dried in order to validate its pouring and to purify the hand. Accordingly, if upon washing for bread one were to switch hands between pourings as one does in the morning, this takes a chance that the impure water of the first pouring of one of the hands will end up getting onto the second hand, such as due to the joint contact of both hands onto the handle of the cup for the sake of pouring the water. This will then invalidate the washing altogether. Accordingly, the Jewish custom which is Torah is to be particular by washing hands for bread to wash each hand consecutively as by doing so one gets rid of the impure water of the first hand before it ever has a chance to get onto the washing cup. Furthermore, we explicitly find in the rulings of Kabbalah of the Arizal, that when washing for bread one is to be particular to wash each hand three times consecutively. All this is in contrast to washing hands in the morning by which on the contrary according to Kabbalah the only way to get rid of the impurity that rests on the hands upon awakening is through washing them inconsecutively, switching off hands between each pouring.
Nonetheless, all this is only the initial custom and Kabbalistic requirement, however, if one did not do so, then there is no intrinsic invalidation of the washing just because one switched off hands between each pouring as one washes in the morning, and it is all dependent on the facts on the ground, and as to whether indeed the impure water of one hand ended up getting onto another hand. Thus, those who wash with a towel as is the Chabad custom, have nothing to worry about as it is not possible for any water of either pouring to ever get onto the vessel and then get onto the other hand. Furthermore, even those who wash without a towel most likely have poured a Revius of water by each pouring onto each hand, by which according to most authorities the water does not contract impurity even by the first pouring, and hence they too have nothing to worry about, and it is only if one poured less than a Revius by the first pouring that there is need to worry. Nonetheless, being that some Poskim, such as Admur in his Seder, suspect for the stringent opinion that even a Revius of water contracts impurity, it is proper to initially be stringent to rewash the hands three times consecutively without a blessing, if there is worry that one of the hands touched the water of the other hand of the first pouring.
Sources: Piskeiy Teshuvos 162:5 See regarding the requirement to wash each hand inconsecutively by the morning washing: Admur Basra 4:2, Siddur; Kama 4:4: “One should not pour on his hands three times consecutively, rather he is to pour once on the right and once on the left and so on and so forth for a total of three times.”; Arizal in Shaar Hakavanos based on Zohar; Shaar Hakavanos of Arizal; Peri Eitz Chaim 3; Matza Shemura 3 [brought in Kaf Hachaim 4:12]; Olas Tamid 17; Beis Oveid 4; Ben Ish Chaiy Toldos 1; See regarding the letter of law allowance to wash the hands inconsecutively even upon washing for bread: Admur 162:13; M”B 162:49 See regarding the Kabbalistic source for washing each hand consecutively upon washing for bread: Kaf Hachaim 162:2; See regarding the tradition of world Jewry to wash each hand consecutively upon washing for bread: Hayom Yom 20th Shevat in name of Rebbe Rashab; Ashkavta Derebbe p. 62 and footnote 40 that it is a tradition dating to the Alter Rebbe to wash each hand three times in a row; Piskeiy Teshuvos 162:5 See regarding the first pouring becoming impure and the worry of that water touching the other hand: Admur in Seder Netilas Yadayim Halacha 2-3; Admur 162:9 and 13; M”B 162:49 See regarding that a Revius of water does not contract impurity although that some are stringent: Seder Netilas Yadayim of Admur Halacha 2 [brings stringent opinion]; Admur 162:1, 9, and 13 [only brings lenient opinion]; M”B 162:49; Stringent opinion: Raavad Tamim Deim 66; Mikvaos 11:3; Rashba Toras Habayis Haruch Bayis Vav Shaar 2, brought in Seder Netilas Yadayim of Admur Halacha 2, M”A 162:3, Shulchan Hatahor; Opinion in M”B 162:21 and Shaar Hatziyon 162:17; Kaf Hachaim See regarding washing each hand in a row as a way of avoiding the water of one hand contacting the other hand: Seder Netilas Yadayim Halacha 2l; Admur 162:13; 158:16; M”A 162:8; Sefer Hateruma 78; Shibulei Haleket 136; Elya Raba 4:5; M”B 162:49