Suppose first that the agent is only indifferent to future Tuesdays, and on the stroke of midnight his preferences change so that he regrets his earlier decision to schedule an agonizing operation for this day. Such fundamental preference changes complicate the case. As Street points out, the different temporal stages of the agent would effectively be at war with each other -- the earlier ones plotting to ensure that the Tuesday-stage suffers agony (supposing this is necessary to spare their later stages, which they care more about, from some lesser pain), and then the Tuesday-stage trying to undo this plot against himself. That is, the Tuesday-agent has a diverging deliberative standpoint, from which he won't endorse or carry out the intentions of his earlier stages. In this sense, it's almost like a new agent temporarily takes over the body each Tuesday, which raises complications regarding whether the prior, Tuesday-indifferent stages are doing something morally objectionable in imposing suffering on the Tuesday stage against "his" will.
To avoid such complications, we may instead suppose the agent is tenselessly Tuesday-indifferent. His reflective preferences remain the same, even on Tuesdays themselves. However, Street argues that when we imagine this agent in vivid detail, he is not so obviously irrational. For during his painful experience, he maintains his "meta-hedonic" indifference to the pain, and so we might think that he achieves a state of emotional calm, and so doesn't really "suffer" from his pain in the ordinary way. (Compare, e.g., a Buddhist monk distancing himself from the searing pain of hot coals.)
However, this scenario too strikes me as containing some potentially confounding complications. In particular, it's no longer clear that the Tuesday experiences we've described are really as painful (hedonically bad) as the experiences felt on other days, and so rather than an agent whose preferences make arbitrary references to Tuesday as such, it seems we've instead described an agent with the more ordinary preference to experiences "pains" on days when they won't cause him suffering (and given his odd constitution, this happens to be Tuesdays).
To clarify this, we need to distinguish two further versions of (tenseless) Tuesday-indifference. Street offers a non-conceptual interpretation, whereby the agent ('Indy') is simply constituted (perhaps due to some bizarre evolutionary story) such that he undergoes regular cycles of psychological transformation. In particular, Indy feels a Buddhist-like 'detachment' from any pain inflicted during special periods (that happen to coincide with Tuesdays), and this indifference naturally carries over to his prospective and retrospective evaluations of "pain" experienced during the special period.
Note that Indy's pain-indifference is prior to his beliefs about what day it is. Locked in a dungeon and deprived of any other temporal cues, he might one day notice a psychological change in himself ("I no longer care about present pain experiences"), and thereby infer that it's Tuesday. This clearly isn't the kind of agent we normally have in mind when talking about Future Tuesday Indifference. Most importantly, Indy's changing psychology corrupts the thought experiment. His phenomenal experience of pain-on-Tuesday is qualitatively different from how he experiences pain on other days, so it could be this qualitative difference, rather than the purely temporal difference, that his preferences are tracking.
So I think it is more worthwhile to consider a psychologically uniform agent with explicitly conceptualized Tuesday-indifference. That is, we should imagine an agent whose psychology is consistent across time, and whose preferences make special reference to "Tuesday" as such (under that description). This means that his beliefs about what day it is will affect his behaviour: in particular, if he falsely believes that today isn't Tuesday, his subjective experiences of pain will feel as agonizing as they do on other days.
This strikes me as the 'pure' version of the thought experiment. After all, to assess Future Tuesday Indifference, we need to hold all else equal, and that means ensuring that the experiences in question are qualitatively identical -- not differing in any phenomenally discernible, and hence potentially hedonically relevant, respect from what he'd suffer on any other day. So let's ask the agent to choose between the following options:
(1) We will wait until Tuesday, and then hook him up to an experience machine that will give him the total phenomenal experience of intense suffering. (This may include brainwashing him into thinking it's Wednesday, or otherwise ensuring that he doesn't "dissociate" himself from the pain in any way that introduces subjective differences.)
(2) We wait until Wednesday, and merely inflict moderate suffering with the experience machine.
The genuinely Future-Tuesday Indifferent agent will choose option (1), by definition. Will Indy?
This poses a dilemma for Street. Indy previously seemed less irrational precisely because we could interpret his Future-Tuesday Indifference as largely compatible with a kind of impartial hedonism: he wasn't really suffering on Tuesdays, after all. But this new choice prises the two motivations apart. He can sanely choose to minimize his suffering, by opting for (2), but then we see that he's not really indifferent to Tuesday suffering after all. (It was just never an issue before, due to his quirky ability to ignore pain on Tuesdays.) Or he can choose option (1), but then we see that his apparent reasonableness was an illusion.
In any case, genuine Future-Tuesday Indifference seems (intuitively) as irrational as ever.