I'm replacing the capacitors in my first radio. Does it matter which direction they're facing? The old ones had obvious markings but I'm not sure about the new ones.

with electrolytics, it definitely matters. Otherwise, I think you may be referring to the old practice of marking the outer foil---that did not have anything to do with polarity---the idea was that a circuit might get better shielding if the outer foil went to ground.** I don't know if any modern caps have this ??

**the theory certainly is plausible, but note that you can always achieve the same thing with a small ceramic or mica that bypasses a physically larger cap. The last time I dabbled in circuit design, this was common practice, but probably mostly for frequencies higher than found in an AM radio

Why not post a scan of the underside so we can see what you are referring to.

If the set has a vertically mounted black cardboard or silver metal cylinder on top of the chassis, advise what the capacitance and working voltage values, printed or embossed on the side, are as well.