I did a quick bisect (of 64bit Windows Chrome)
OK in 43.0.2357.81, released 5/25/15  (8 spokes)
broken in 44.0.2403.89, stable released 7/21/15 (6 spokes)
   46.0.2490.80, stable released 10/22/15

This looks like a bug in the Segoe UI Symbol font, which shows this character as a six-pointed wheel: http://www.fileformat.info/info/unicode/char/2638/fontsupport.htm

Repros on Windows 10, not on Linux.

Logged https://developer.microsoft.com/en-us/microsoft-edge/platform/issues/8143689/ . Edge isn't really the right component to report this to, but I can't figure out how to log a font bug with Microsoft, and Edge can probably do more to fix this than we can.

The Local Font List tool
  http://www.fileformat.info/info/unicode/font/fontlist.htm?text=%E2%98%B8+-+Unicode+Character+%27WHEEL+OF+DHARMA%27+%28U%2B2638%29
(embedded in the test URL page in the CR description)
shows nearly all the local fonts using 6 spokes.  
Only 4 or 5 MS* fonts, and Arial Unicode MS, use 8 spokes.
The broken Segoe UI Symbol font is in good company.

The local font list displays the same in Chrome and IEx, for Windows.
Font lists vary by OS. 

The local font list displays exactly the same glyph for nearly all of those fonts. Instead, the character map indicates that those fonts do not actually contain a glyph for that character. The likely reason for that is that the local fonts list tool is using font fallback to pull the glyph from a different font (Segoe UI Symbol, in this case) for those fonts that do not contain the glyph. Not exactly the ideal behavior for that tool, but that's not something we can change.

@kulshin: Yes, probably font fallback in the local list tool, which had crossed my mind too.  Oh well..

Is there a complementary mechanism to prioritize and take U+2638 from one of the local 8 spoke fonts, instead of Segoi UI Symbol?

We could in theory override font fallback for a single character (we currently do it on a character-block level for some character blocks), assuming we had a good list of fallback fonts to use. However, unless we get multiple requests for that fix, I think it's better to encourage Microsoft to fix their font.

Agreed with comment 2 and 8. This should be filed as a font bug.

Sorry for the chatter.  Just now discovered kulshin logged this as Microsoft (Edge) bug 8143689 yesterday.

Updated MS 8143689 to refer back to here.  It's marked there as impacting Edge, IE and Chrome.

Although this is a Segoe UI Symbol font bug, something changed in Chrome at m44 to expose the problem.  On Win8.1, with todays MS fonts, 43.0.2357.81 displays U+2638 correctly with 8 spokes for every font except Segoe UI Symbol, using the local font list tool:
  http://www.fileformat.info/info/unicode/font/fontlist.htm?text=%E2%98%B8+-+WHEEL+OF+DHARMA+%28U%2B2638%29

  

Can you post a screenshot of the local font list tool in Chrome M43?

I'm not super familiar with the history of font fallback and font rendering in Chrome, but it's possible we changed our fallback algorithm around that time to select a different fallback font, now favoring Segoe UI Symbol.

Generated with 43.0.2357.81 with
  http://www.fileformat.info/info/unicode/font/fontlist.htm?text=%E2%98%B8+-+WHEEL+OF+DHARMA+%28U%2B2638%29
The tool apparently uses the flashplayer (which griped: stale..). I used the m43 bundled 17.0.0.188

I pieced together sections with Arial Unicode MS, MS*, and Segoe UI fonts.
All are 8 spoke, some are certainly fallback.

If you need anything else from m43, it's easy for me to pull up.

Deleted: Dharma-Chr43..81-FontList.png 177 KB

	Dharma-Chr43..81-FontList.png 177 KB View Download

FYI: Segoe UI Symbol has been replaced by Segoe MDL2 Assets in Win10
  https://msdn.microsoft.com/windows/uwp/style/segoe-ui-symbol-font
I'm not sure in what contexts, mostly for icons. The article reads..
  The Windows 8/8.1 Segoe UI Symbol icon font has been replaced with the Segoe MDL2 Assets font effective with the release of Windows 10.
...
  Segoe UI Symbol will still be available as a "legacy" resource
...
  Unlike with Segoe UI Symbol, the icons in the Segoe MDL2 Assets font are not intended for use in-line with text. 

Ilya: Can you check if/how U+2638 displays in MDL2 (only on Win10)?

Since the MS bug 8143689 is against Segoe UI Symbol, MS may opt to use MDL2?

Based on http://modernicons.io/segoe-mdl2/cheatsheet/ it appears Segoe MDL2 Assets does not contain a glyph for U+2638. Further, it appears that font encodes codepoints in the private use area, so I would not count on it encoding any particular character.