Let's say, for instance, I cast Sakashima's Protege and cascade into Jorn, God of Winter. Do I get to choose which side it copies?
Furthermore, if I have a kaldheim god in play and cast Clever Impersonator, do I chose which side it copies?
I'm just trying to understand how cloning and modal cards work.
For your first question: Yes, but in an indirect way. You choose which face (not: side) of the card Jorn, God of Winter to cast with cascade (either the front face Jorn, God of Winter, or the back face Kaldring, the Rimestaff)** (C.R. 712.7, 702.85a). (Note that both faces each have mana value less than Sakashima's Protege's of 6 [review C.R. 702.85a].) Say you choose the Kaldring face. Then the spell will appear as Kaldring and get to enter the battlefield with that face up by default, namely the Kaldring face, before Sakashima's Protege itself resolves (C.R. 712.9). Then when Sakashima's Protege resolves, it enters the battlefield as a copy of Kaldring, if you choose (but you can't choose the Jorn face instead) (C.R. 712.4e).
For your second question: No. A modal double-faced permanent "has only the characteristics of the face that's up" (C.R. 712.4e), so that if a Jorn is on the battlefield and you choose that permanent this way, you can't have Clever Impersonator enter as a copy of its other face, namely Kaldring, instead.
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Note that in both cases, if an object (such as Sakashima's Protege or Clever Impersonator) becomes a copy of a modal double-faced permanent (such as Jorn or Kaldring), it becomes a copy of whatever face that permanent shows (there is no choice to copy the other face of that permanent instead) (C.R. 712.4e, 707.2).
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** You can choose to cast the front or back face this way with cascade but only if the "resulting spell's mana value is less than" that of the spell with cascade (C.R. 712.7, 702.85a; for a related situation with split cards, see also this thread).
EDIT (Feb. 16): Edited in view of recent rule changes.
EDIT (Mar. 7): Edited to conform to the text as it actually appears in the comprehensive rules.
EDIT (Sep. 21, 2021): Edited, including because some rules were renumbered in the meantime.
EDIT (Jun. 17, 2022): Edited, including to use updated terminology.
Furthermore, if I have a kaldheim god in play and cast Clever Impersonator, do I chose which side it copies?
I'm just trying to understand how cloning and modal cards work.
For your second question: No. A modal double-faced permanent "has only the characteristics of the face that's up" (C.R. 712.4e), so that if a Jorn is on the battlefield and you choose that permanent this way, you can't have Clever Impersonator enter as a copy of its other face, namely Kaldring, instead.
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Note that in both cases, if an object (such as Sakashima's Protege or Clever Impersonator) becomes a copy of a modal double-faced permanent (such as Jorn or Kaldring), it becomes a copy of whatever face that permanent shows (there is no choice to copy the other face of that permanent instead) (C.R. 712.4e, 707.2).
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** You can choose to cast the front or back face this way with cascade but only if the "resulting spell's mana value is less than" that of the spell with cascade (C.R. 712.7, 702.85a; for a related situation with split cards, see also this thread).
EDIT (Feb. 16): Edited in view of recent rule changes.
EDIT (Mar. 7): Edited to conform to the text as it actually appears in the comprehensive rules.
EDIT (Sep. 21, 2021): Edited, including because some rules were renumbered in the meantime.
EDIT (Jun. 17, 2022): Edited, including to use updated terminology.