I wondered this a while ago as the IOS images for the 3560E/3560X (and 3750E/3750X) are the same up to 15.0(1)SE12. Its difficult to see now as they 3560E/3750E are EoL/EoS so don't appear on CCO any longer. The 3560X/3750X are likely to disappear shortly as they went EoS last month.
I'd say try it and see what happens? You should be able to recover from the console/OoB Ethernet interface - just make sure you have the .tar image for 15.0(1)SE12 available.
The C3560X 15.2(4)E10 .tar file has some 3560E .gif images in the /HTML/images folder, however it also has 3560G images in there so the HTML might be universal for all 3560 models.
The info file in the .tar file also has this:
version_suffix: universalk9-152-4.E10
version_directory: c3560e-universalk9-mz.152-4.E10
image_system_type_id: 0x00000002
image_name: c3560e-universalk9-mz.152-4.E10.bin
ios_image_file_size: 26778112
total_image_file_size: 32061952
image_feature: IP|LAYER_3|PLUS|SSH|3DES|MIN_DRAM_MEG=128
image_family: C3560E
stacking_number: 1.59
board_ids: 0x0000006c 0x0000006d 0x0000006e 0x0000006f 0x00000067 0x000000b9 0x0000021b 0x000000b5 0x000000b6 0x0000021c 0x000000b7 0x000000b8 0x000000e6 0x000000e5 0x000000e2 0x000000e1 0x000000e9 0x000000e8 0x00000240 0x00000241 0x00000242
fru_module_version: 03.08.10.WAL
info_end:
